Dankelmann-The Distance Concept and Distances in Graphs

Home , Distance (graph theory), Lollipop graph

The Distance Concept and Distance in Graphs

P. Dankelmann and S. Mukwembi University of KwaZulu-Natal, Durban, South Africa.

November 2, 2011

Contents

1 Introduction 2

2 Diameter 3 2.1 Diameter, order and size ...... 3 2.2 Diameter and degrees ...... 4 2.3 The degree diameter problem ...... 5 2.4 Diameter, connectivity and edge-connectivity ...... 7 2.5 Diameter-minimal graphs ...... 7 2.6 Remarks on computational aspects ...... 9

3 Radius 9 3.1 Radius, diameter and the centre ...... 10 3.2 Radius, size, and vertex degrees ...... 11 3.3 Connectivity and edge-connectivity ...... 13 3.4 Planar graphs ...... 14 3.5 Domination number and independence number ...... 14 3.6 Radius-critical and minimal graphs ...... 14

4 Eccentricity 16

5 Wiener index 19

6 Eccentric connectivity index 22 6.1 Some extremal graphs ...... 24 6.2 Bounds in terms of order ...... 24 6.3 Bounds in terms of order and diameter ...... 26 6.4 Bounds in terms of order and size ...... 27 6.5 Regular graphs ...... 28

7 Degree distance 29

8 Gutman index 32

9 Eccentric Distance Sum 34

1 1 Introduction

The concept of distance pervades all of science and mathematics, and even our daily lives. Also in the study of graphs, distances have played a central role throughout. A major impetus to investigations of distance concepts in graphs was given by their wide applicability. Its applications range from facility location problems and network design in operations research to prediction of properties of chemical compounds in chemistry, from measuring closeness of groups of individuals in sociology to identifying important role players in, for example, the internet. With Wiener’s discovery of a close correlation between the boiling points of certain alkanes and the sum of the distances between vertices in the graphs representing their molecular structures [163], it became apparent that graph parameters, or topological indices, can potentially be used to predict properties of chemical compounds. Many new topological indices have been considered over the past decades and their predictive power for various properties tested. Like Wiener’s original topological index, which is defined as the sum of all distances, many topological indices are based on distances between vertices. In the classical study of distances in graph theory, the main focus has been on the study of the two main graph parameters concerned with distance, the diameter and the radius. Methods developed became the foundation for most research into other distance related graph parameters. For this reason, we briefly review some of the main results on these two classical distance parameters, before moving on to the Wiener index and more recent topological indices. Although most results will be presented without proof, we include some of the proofs where methods are used that seem applicable to other distance based topological indices. The notation we use is as follows. The distance between two vertices u and v of a graph is the minimum length of a (u, v)-path. It is denoted by dG(u, v). The eccentricity ecG(v) of a vertex v is defined as the distance between v and a vertex furthest apart from v, i.e., ecG(v) = maxu∈V (G) dG(v, u). The largest of the eccentricities of the vertices of a graph G (or alternatively the largest of the distances between its vertices) is the diameter, denoted by diam(G). The radius of G, denoted by rad(G), is the smallest of the eccentricities of the vertices. 2 Diameter

The diameter is the most common of the classical distance parameters in graph theory, and much of the research on distances is in fact on the diameter.

2.1 Diameter, order and size

Let G be a connected graph of order n. Clearly, 1 ≤ diam(G) ≤ n − 1, and the diameter equals 1 or n − 1 if and only if G is a complete graph or a path. If we also consider the size of the graph, then we can give an upper bound on the diameter which is signiﬁcantly stronger than n − 1: 1 r 17 diam(G) ≤ n + − 2m − 2n + . (1) 2 4 This bound is an immediate consequence of a classical result (Theorem 1) by Ore [140] which characterises diameter-maximal graphs, i.e., graphs with the property that adding any new edge decreases the diameter, which leads to the determination of the maximum size of a graph of given order and diameter.

Theorem 1. Let G be a connected graph of order n and diameter d ≥ 2. (a) G is diameter-maximal if and only if

G = K1 + Kn1 + Kn2 + ... + Knd−1 + K1 for some positive integers n1, n2, . . . , nd−1 with n1 + n2 + ... + nd−1 = n − 2. (b) If G has size m, then 1 m ≤ d + (n − d − 1)(n − d + 4). 2 Part (a) of the above theorem is proved by ﬁxing a pair u, v of vertices at distance d, and then considering the ’ith distance layer’ of u for i = 1, 2, . . . , d, i.e., the set of vertices at distance exactly i from u. It is easy to check that adding an edge between two vertices in the same or in consecutive distance layers does not decrease d(u, v), and so leaves the diameter unchanged, hence all such edges must be present in G. If now ni is the cardinality of the ith distance layer, then it is easily seen that nd = 1. Part (b) now follows from the fact that the size of a graph described in (a) is maximised if, for example, all but one ni (for i 6= d) equal 1. QED

Several extensions of the above theorem are known, where in addition to order and diameter also other graph parameters such as connectivity or edge-connectivity are prescribed, and the maximum number of edges of such is determined. For these the reader is referred to papers by Caccetta and Smyth [26, 27, 28]. It is also easy to prove a bipartite version of Theorem 1. Lower bounds for the diameter in terms of order and size alone are not particularly interesting since every connected graph of order n has at least n − 1 edges, and since for each n and each d with 2 ≤ d ≤ n − 1 there exists a tree (of size n − 1) of order n and diameter d. However, for small diameter such a tree has large maximum degree, and its minimum degree equals 1. It makes therefore sense to consider relationships between order, size and diameter that take also maximum degree or minimum degree into account. The challenging problem of determining the minimum number of edges of a graph of given order, maximum degree and diameter has been investigated by Erdösand Rényi [68], Erdös, Rényi and Sós[69], and by Bollobás[15]. See also the book [18]. The problem of determining the minimum number of edges of a graph of given order, minimum degree and diameter (see Bondy and Murty [19]) was considered by Bollobás and Harary [24], and solved by Bollobás[17].

2.2 Diameter and degrees

Since the graph that maximises the diameter among all connected graphs, the path, has minimum degree 1 and maximum degree 2, it is natural to ask if for graphs with larger maximum degree or larger minimum degree we can give stronger bounds. It is straightforward to prove that a graph of order n and maximum degree ∆ has diameter at most n + 1 − ∆, which was first noted in [20]. Upper bounds on the diameter in terms of order and minimum degree are far more interesting, they have been considered and rediscovered by numerous authors, for example Moon [127], Goldsmith, Manvel and Farber [83], Erdös,Pach, Pollack and Tuza [67]. Klee and Quaife [110] and Amar, Fournier and Germa [3] proved similar bounds that also prescribed the vertex connectivity. Below we give the most general result, due to Erdöset al. [67]. The basic observation from which most of these bounds follow is simple: If we fix a shortest path between two vertices u and v at maximum distance and if we then consider the vertices on the path at distance 0, 3, 6, 9,... on this path, then we obtain about diam(G)/3 vertices whose closed diam(G) neighbourhoods are pairwise disjoint. Hence, approximately, n ≥ 3 (δ + 1), and so the 3n diameter is at most δ+1 + O(1).

Theorem 2. Let G be a connected graph of order n and minimum degree δ ≥ 2. Then

j 3n k diam(G) ≤ . δ + 1 If G is also triangle-free then ln − δ − 1m diam(G) ≤ 4 2δ If G does not contain 4-cycles, then

j 5n k diam(G) ≤ δ2 − 2bδ/2c + 1

In [67] it is also shown that the first two bounds are best possible, up to an additive constant, and that the third bound is almost best possible. Dankelmann, Dlamini and Swart [41, 42] used different methods to extend Theorem 2 to K2,t-free graphs and to K3,3-free graphs. It makes sense to investigate if bounds similar to those for graphs of given order and minimum degree hold, with the minimum degree replaced by a number dependent on the degree but slightly larger than the minimum degree, for example one could try the average degree. It is easy to verify that replacing the minimum degree by the average degree in Theorem 2 yields an inequality which does not hold in general. Another promising candidate seems to be the inverse degree of a graph, defined by r(G) = P 1 , as r(G) equals n , where dîs the harmonic mean of the v∈V (G) deg(v) dˆ vertex degrees of G. This might have been one of the underlying reasons for a conjecture by the computer programme GRAFFITTI [72, 73] that the (arithmetic) mean of the distances between all pairs of vertices in a graph is at most its inverse degree. In response to this conjecture, Erdös, Pach and Spencer [66] constructed an infinite class of graphs whose diameter is at least

r(G) log n 2b c + o(1) , 3 log log n thus refuting the conjecture. They also showed that always

log n diam(G) ≤ 6r(G) + 2 + o(1) . log log n

log n So the maximum diameter of a graph in terms of order and inverse degree is O(r(G) log log n ). The gap between the upper and the lower bound was reduced by Dankelmann, Swart and van den Berg [45], who improved the upper bound by a factor of about 2. Mukwembi [131] considered relationships between the inverse degree and the diameter for trees, planar graphs, and for chemical graphs, i.e., graphs of maximum degree at most 4.

2.3 The degree diameter problem

As pointed out above, it is easy to construct graphs of large order but small diameter, the star

K1,n−1 being the prime example. However, if practical considerations dictate that no vertex should have degree greater than, say, ∆, then restrictions arise. Let G be a connected graph of maximum degree ∆ and diameter d. If we ﬁx a vertex v and consider the distance layers of v, then clearly there are at most ∆ vertices a distance 1 from v. Each vertex in the ﬁrst distance layers is adjacent to at most ∆ − 1 vertices in the next distance layer, so there are at most ∆(∆ − 1) vertices at distance 2 from v. Similarly there are at most ∆(∆ − 1)2 vertices at distance 3 and so on. Adding these upper bounds for all distance layers, we get that

(∆ − 1)d − 1 n ≤ 1 + ∆ + ∆(∆ − 1) + ∆(∆ − 1)2 + ... + ∆(∆ − 1)d−1 = 1 + ∆ . (2) ∆ − 2

This bound on the order of a graph in terms of diameter and maximum degree is called the Moore bound, and graphs which attain this bound are called Moore graphs. It turns out that there are no Moore graphs of diameter greater than 2 other than odd cycles. For diameter 2, however, Moore graphs exist: the 5-cycle for ∆ = 2, the Petersen graph for ∆ = 3, and for ∆ = 7 the Hoﬀman-Singleton graph. There is no other Moore graph of diameter 2 except possibly of degree ∆ = 57. This is shown by considering the eigenvalues of the adjacency matrix, and a proof can be found in many standard texts on graph theory. The existence of a Moore graph with ∆ = 57, and hence with 3250 vertices, is one of the great outstanding problems in the theory of distances.

Deﬁne n∆,d to be the maximum order of a graph of diameter d and maximum degree at most

∆. The determination of the values of n∆,d is the so-called degree-diameter problem, sometimes simply called the (∆,D)-problem, which has a considerable literature. Denote the Moore bound by M∆,d. Since there are only few Moore graphs, n∆,d ≤ M∆,d − 1 for all values of ∆ and d with the above-mentioned exceptions. Bannai and Ito [9] managed to improve this bound to n∆,d ≤ M∆,d − 2 for all d ≥ 3, and for some small values of ∆ there are slightly better bounds, but all only by small additive constants. For example Miller and Pineda-Villavicencio [123] proved n3,d ≤ M3,d − 6 for all d ≥ 5. This is in stark contrast to the currently best lower bound on n∆,d for large values of ∆ and d: a construction by Canale and Gómez[30] shows that for sufficiently large values of d which are congruent 0, 1 or 7 mod 8, and infinitely many values

∆ d of ∆ we have n∆,d ≥ ( 1.45 ) . A good starting point for the literature on the degree-diameter problem is the dynamic survey by Miller and Sir´aˇn[122].ˇ 2.4 Diameter, connectivity and edge-connectivity

No signiﬁcant improvement of the above bounds can be obtained if above the minimum degree is replaced by the edge-connectivity and if the edge-connectivity is not too small. Clearly, since λ(G) ≤ δ(G), we obtain the bounds in Theorem 2, only with δ(G) replaced by λ(G). For example the ﬁrst bound in Theorem 2 becomes

j 3n k diam(G) ≤ . λ + 1

If λ(G) ≥ 8 then it is easy to construct graphs whose diameter is close to the bound given above. For example if λ(G) ≡ 2 (mod 3) the graph Kλ−1 + K(λ+1)/3 + K(λ+1)/3 + K(λ+1)/3 +

... + K(λ+1)/3 + Kλ are clearly λ-edge-connected and their order diﬀers from the above bound only by an additive constant. For smaller values of λ it is easy to see that one can improve the

3n bound λ(G)+1 + O(1). If we consider the diameter of k-connected graphs rather than k-edge-connected graphs or graphs of minimum degree k, then we obtain a stronger bound on the diameter. This follows directly from the fact that by Menger’s Theorem in a k-connected graph there exist k internally vertex-disjoint paths between any two vertices that the diameter of a k-connected graph is at

n−2 most 1+b k c. This bound is sharp as can be seen from the graph K1 +Kk +Kk +...+Kk +K1. The diameter of graphs that are k-connected and also k-regular were considered by Klee and Quaife [111] and by Myers [132, 133].

2.5 Diameter-minimal graphs

Various concepts of minimality and criticality with respect to the diameter have been investigated. Unfortunately the terminology is by no means uniform throughout the literature. We ﬁrst consider criticality with respect to edges. Following the terminology of the classic book on distance in graphs by Buckley and Harary [24], we call a connected graph G diameter- minimal if deletion of any edge increases the diameter, diameter-maximal if addition of any edge not in G decreases the diameter. The most important result on diameter-maximal graphs, Theorem 1, has been stated above, so we focus on diameter-minimal graphs. Examples of a diameter-minimal graphs of diameter 2 are the complete bipartite graphs. Plesnik [144] showed

3 that a diameter-minimal graph of order n has at most 8 n(n − 1) edges. This bound, however, is not best possible. A famous conjecture by Simon and Murty (see for example [25]) says that

n2 for every n ≥ 3 a diameter-minimal graph of order n and diameter 2 has at most b 4 c edges, and that the balanced bipartite graph of order n is the unique extremal graph. For several decades this conjecture has defied attempts to prove it. Caccetta and Häggkvist[25] proved that a diameter-minimal graph of diameter 2 has at most 0.27n2 edges, a result very close to the conjecture. Fan [74] proved the even stronger upper bound 0.2532n2. Füredi[77] showed that the conjecture holds for all values of n greater than a given value n0, which is very large (approximately a tower of 2s of height about 1014). Also for graphs of diameter greater than 2 we do not know the maximum number of edges in a diameter-minimal graph. A conjecture by Caccetta and Häggkvist[25] on the maximum number of edges of a diameter-minimal graph was refuted by Krishnamoorthy and Nandakumar [114]. We now consider the effect of removing a vertex on the diameter. Clearly, if G is a graph and v a vertex of G, then the diameter of G−v can either increase (for example if G is a cycle of length at least 5), decrease (for example if G is a path and v an end vertex) or remain unchanged. Therefore three concepts of criticality with respect to vertex removal appear natural: removal of any vertex increases the diameter, removal of any vertex decreases the diameter, and removal of any vertex changes the diameter. Clearly, if u and v are two vertices at distance equal to the diameter in G, then removing a vertex in V (G) − {u, v} cannot decrease their distance. Hence every connected graph has at most two vertices whose removal decreases its diameter, as observed by Gliviak [80]. The only connected graph with the property that the removal of any vertex decreases its diameter is therefore K2, and so this type of diameter criticality is not particularly interesting. Graphs for which removal of any vertex increases the diameter are far more interesting. In the more recent literature, these graphs have been called diameter-critical graphs, and we adopt this terminology here. Examples of diameter-critical graphs are cycles of length at least 5 and the Petersen graph. Plesnik [144] conjectured that diameter-critical graphs exist for every degree and every diameter. This was proved 24 years later by Caccetta and El-Batanouny [29]. The currently smallest such graphs were constructed by Royle [147]. Plesnik [144] showed that every diameter-critical graph is 2-connected. The question by how much the diameter can increase if a vertex is removed from a diameter-critical graph G was considered by Boals et al. [13]. The showed that if diam(G) = 2, then diam(G − v) = 3, and if diam(G) = 3 then diam(G − v) ∈ {4, 5}. They conjectured that if G is a diameter-minimal graph of diameter k then diam(G − v) ≤ 2k − 1 for every vertex v of G. For diameter-minimal graph of diameter 2 they also conjecture that no vertex has degree greater than |V (G)|/2. Boals et al. [13] showed that every graph is an induced subgraph of a diameter-minimal graph. The third concept of vertex-criticality, removal of any vertex changes the diameter, was investigated by Gliviak [80], who showed in particular that every graph is an induced subgraph of such a graph. Vacek [157] considered graphs in which addition of any edge (or deletion of any edge, or deletion of any vertex) leaves the diameter unchanged. The most extreme graph in this sense is a graph in which one can remove edges without increasing the diameter until only a spanning tree is left. Graphs that have a diameter-preserving spanning tree have been characterised by Buckley and Lewinter [23] as graphs that satisfy either (i) diam(G) = 2rad(G), or (ii) G has a pair of adjacent central vertices with no common neighbour and diam(G) = 2rad(G) − 1.

2.6 Remarks on computational aspects

We briefly consider the problem of how distances can be computed efficiently. The distance between two vertices is easily computed using a technique known as breadth first search. It computes the distances from a root vertex to all other vertices, and it yields a spanning tree of the graph which preserves the distances from the root. The fact that for every vertex v in a graph there is a spanning tree that preserves the distances from v is used frequently. The running time of breadth-first-search is O(|E|) if adjacency lists are used, so the algorithm is n very efficient. Since the size of every graph is bounded by 2 , one can compute the distances between all pairs of vertices in time O(n3), and this can be used to compute most of the distance based topological indices for individual graphs in time O(n3). For graph classes in which the size is bounded linearly in the order, which is the case for trees, graphs whose maximum degree is bounded by a constant (such as chemical graphs), planar graphs, unicyclic graphs, one can compute the distances between a given vertex and all other vertices in time O(n). Hence the distances between all pairs of vertices, and thus many distance based topological indices, can be computed in time O(n2) for such graphs. There are also other graph classes, such as interval graphs for which this is likely to be the case.

3 Radius

The radius is, after the diameter, the second most important classical distance parameter in graph theory. Although it is more useful for facility location problems in Operations Research than in Chemical Graph Theory, many results on the radius and the techniques applied to proving it can be applied to distance based topological indices. The centre of a connected graph is the set of vertices whose eccentricity equals the radius of the graph, and the graph induced by the centre is the central subgraph.

3.1 Radius, diameter and the centre

We begin by brieﬂy considering the relationship between the radius and the diameter. A very useful basic inequality follows directly from the deﬁnition of the radius and from the triangle inequality: rad(G) ≤ diam(G) ≤ 2rad(G). (3)

For trees we have a much stronger relationship between the radius and the diameter, they almost determine each other, as shown by the following classical theorem which is essentially due to Jordan [108]. It is the earliest result on the structure of central subgraphs.

Theorem 3. Let T be a tree of order n ≥ 2. (a) The centre of T consist of a single vertex or of two adjacent vertices. (b) If the centre of T consists of a single vertex then diam(T ) = 2rad(T ), and if the centre of T consists of two adjacent vertices then diam(T ) = 2rad(T ) − 1.

Proof. The proof is by induction on the radius. The trees of radius 1 and 2 are the stars and double brooms, respectively, and it is easy to verify the theorem for those. If rad(T ) ≥ 3, then consider the tree T 0 obtained from T by removing all end vertices. Since removing the end vertices reduces the radius by 1, the diameter by 2, and leaves the centre unchanged, the theorem follows. QED Harary and Norman [91] generalises Jordan’s theorem by showing that the centre of a graph G is contained in a block of G. In view of Theorem 3, one might expect that also strong statements can be made about the structure of central subgraphs. This is unfortunately not the case. A folklore result on centres states that every graph G is isomorphic to the central subgraph of some graph H. The graph H can be constructed from G by adding two new vertices, u and v, which are adjacent to every vertex of G but not to one another, and then adding two further vertices u0 and v0 which are adjacent only to u and v, respectively. It is easy to verify that G is the central subgraph of H since only the vertices in G have eccentricity 2. A consequence of the result that every graph is isomorphic to the central subgraph of some graph is that no statement can be made about properties of central subgraphs, except for statements that hold for all graphs anyway. However, for particular graph classes, one can make statements about the structure of their central subgraphs. Laskar and Shier [115] showed that the central subgraph of a chordal graph is always connected. Indeed, Yeh and Chang [165] report that Soltan and Chepoi [150] have shown that its diameter cannot exceed 3. For outerplanar graphs the structure of the center is even more restrictive, it can be only one of seven graphs, as shown by Proskurowski [146]. There are several further results in the literature on centres of graphs belonging to various graph classes. For such graph classes one can often prove relationships between the radius and diameter stronger than (3). For example for chordal graphs, Laskar and Shier [115] showed that, if a chordal graph has radius r and diameter d, then d/2 ≤ r ≤ bd/2c + 1. A good starting point for reading in this area is the paper by Yeh and Chang [165] on centres and medians of distance-hereditary graphs. It is easy to construct graphs with given radius r and diameter d, as long as r and d are positive integers satisfying (3). The smallest order of such graphs was determined by Ostrand [141]. An important technique of proving upper bounds on the radius for connected graphs is to prove ﬁrst that it holds for all trees, and then to make use of the fact that the radius of a connected graph is not greater than the radius of any of its spanning trees. This technique is less applicable for proving bounds on the diameter because every connected graph has as spanning tree that preserves the radius, but not every connected graph has spanning tree that preserves the diameter. For example Theorem 3 implies that the radius of a tree of order n is

n at most 2 . We immediately get the following corollary.

Corollary 1. Let G be a connected graph of order n. Then

n rad(G) ≤ . 2

3.2 Radius, size, and vertex degrees

It is natural to expect that a connected graph G whose size is large relative to the order of G has small radius. So it is natural to look for an upper bound on the radius in terms of order and size. Vizing [160] solved the equivalent problem of determining the maximum size of a graph of given order and radius. Theorem 4. Let G be a graph of order n and radius r. Then   n(n − 1)/2 if r = 1, m(G) ≤ n(n − 2)/2 if r = 2,  (n2 − 4nr + 5n + 4r2 − 6r)/2 if r ≥ 3.

The above bound is sharp for all n and r. For r = 1 and r = 2 equality if attained by a complete graph and a complete graph with the edges of a perfect matching or a minimum edge cover removed. For r ≥ 3 equality is attained by the graph obtained from the union of C2r and

Kn−2r by choosing three consecutive vertices on the cycle and joining these three vertices to all vertices of the complete graph. Note that these are not the only extremal graphs. The maximum number of edges in a bipartite graph of given order and radius was determined by Dankelmann, Swart and van den Berg [46]. The minimum size of a connected graph of given order n and radius is of course n − 1. The smallest size of graphs of prescribed order, radius and minimum degree was determined by Dankelmann and Volkmann [50]. We now bound the radius in terms of order and vertex degrees, starting with the maximum degree. The sharp bound n − ∆ + 2 rad(G) ≤ 2 is easily proved by induction on n for trees, and then extended for all connected graphs by considering a spanning tree with at least ∆ end vertices. Finding bounds on the radius in terms of minimum degree is more diﬃcult. The above bounds (Theorem 2) on the diameter in terms of order and minimum degree have analogues for the radius. Erd¨os,Pach, Pollack, and Tuza [67] proved the following bounds, which are sharp up to an additive constant.

Theorem 5. Let G be a connected graph of order n and minimum degree δ ≥ 2. Then

3(n − 3) rad(G) ≤ + 5. 2(δ + 1)

If G is also triangle-free then n − 2 rad(G) ≤ + 12. δ If G does not contain 4-cycles, then

j 5n k rad(G) ≤ 2(δ2 − 2bδ/2c + 1)

The proof of Theorem 5 is based on the observation that, given a centre vertex v, there exists vertices wi at distance r or r − 1 from v, i = 1, 2, and shortest paths Pi from v to wi,with the following property: no vertex u1 of P1 shares a neighbour with a vertex u2 of P2, unless u1 or u2 are very close to one of the vertices v, w1 or w2. Given P1 and P2 one can ﬁnd approximately

2rad(G)/3 vertices with disjoint neighbourhoods by choosing every third vertex on P1 and P2. 2 This yields, approximately n ≥ 3 rad(G)(δ + 1), and so the ﬁrst bound. QED With diﬀerent methods, Dankelmann, Dlamini and Swart [41, 42] proved similar bounds for

K2,t-free graphs and K3,3-free graphs.

3.3 Connectivity and edge-connectivity

We ﬁrst bound the radius in terms of order and edge-connectivity. As for the diameter, for most values of λ(G) we can use Theorem 5 and the fact that the minimum degree of a graph is greater or equal to its edge-connectivity to obtain sharp bounds on the radius for most values of λ(G). The only value of λ(G) for which the bound thus obtained is not sharp is λ(G) = 3, for which

3 Theorem 5 yields only the bound rad(G) ≤ 8 n + O(1). A best possible upper bound is given in the following theorem by Dankelmann, Mukwembi and Swart [44].

Theorem 6. Let G be a 3-edge-connected graph of order n. Then

n + 17 rad(G) ≤ . 3

Apart from an additive constant, this bound is best possible.

Mukwembi [130] showed that, if the graph is also bipartite, then this bound can be improved

3n+112 to rad(G) ≤ 10 . We now turn to the vertex-connectivity. It follows directly from Menger’s Theorem that

n−2 every k-connected graph of order n has diameter at most 1 + b k c. Since rad(G) ≤ diam(G), this is also an upper bound on the radius of a k-connected graph. That it is sharp for even k

k is seen from the graph 2 th power of the cycle Cn. Determining a sharp upper bound for the radius of a k-connected graph for odd k is much harder. Harant and Walther [90] proved that

n n for 3-connected graphs rad(G) ≤ 4 + O(log n) and conjectured that rad(G) ≤ 4 + C for some constant C. This was proved by Harant [89]. A slight improvement in the additive constant was given by Iida [101]. Egawa and Inoue [62] generalised these results for all odd k.

Theorem 7. Let k be an odd integer, and let G be a k-connected graph of order n. Then

n + k + 10 rad(G) ≤ . k + 1 This bound is best possible, up to an additive constant. This can be seen by considering the

k+1 ( 2 )th power of the cycle Cn, which also is an example for sharpness of the bound for (k + 1)- connected graphs. So the bound on the radius of k-connected graphs, for odd k, is essentially the same as for (k + 1)-connected graphs. We note that a bound similar to that by Egawa and Inoue, but with a slightly better additive constant, was proved by Iida and Kobayashi [102].

3.4 Planar graphs

A lower bound on the radius was given by Kim and West [113], who showed that the radius of a triangle-free planar graph with no vertex of degree 1 or 2 is at least 3. Harant [88] gave a bound on the radius of a 3-connected planar graph in terms of order and maximum face length. This bound was improved by Ali, Dankelmann and Mukwembi [2]:

Theorem 8. Let G be a 3-connected plane graph on n vertices with maximum face length `. Then n 5` 5 rad(G) ≤ + + . 6 6 6 For constant `, this bound is best possible apart from the additive constant.

In [2] a similar bound for 4-connected planar graphs is also given.

3.5 Domination number and independence number

3 Delavi˜na,Pepper and Waller [54] showed that rad(G) ≤ 2 γ(G). Henning and Mukwembi [98] showed that a graph of order n, radius r, domination number γ and minimum degree δ satisﬁes

4 2 γ ≤ n − 3 r + 3 , and if G has minimum degree at least 3, then this can be improved to γ(G) ≤ 2 n − 3 (r − 6)δ, thus obtaining new bounds on the radius in terms of order, domination number and minimum degree. The computer programme GRAFFITI produced several conjectures on the radius. One of the most popular conjectures, for which several proofs were found (see for example [52, 70]), is that the radius of a graph is at most its independence number. Moreover, the computer programme conjectured that a graph with rad(G) = α(G) has a hamiltonian path. This was proved by Delavi˜na,Pepper and Waller [53].

3.6 Radius-critical and minimal graphs

Various concepts of criticality, in the sense that no vertex or edge can be deleted without changing the radius, have been investigated. The graphs for which removal of any edge either increases the radius or disconnects the graph are exactly the trees, as observed by Gliviak [79]. This follows directly from the fact that every connected graph has a radius-preserving spanning tree. Following the terminology in [24], a graph with the property that adding any edge decreases the radius is called radius-maximal. Radius-maximal graphs have been studied by Nishanov [136], Harary and Thomassen [92], and Gliviak, Knor and Soltés[81,ˇ 82] but no characterisation is known. This is in marked contrast to the situation for graphs where addition of any edge decreases the diameter, which have a simple characterisation. In [159], Vacek poses the problem of determining the maximum size of a radius-maximal graph of given order and radius. Graphs with the property that deletion of any edge does not increase the radius are called radius-edge-invariant graphs. These are considered by Walikar, Buckley and Itagi [161] and by Bálint and Vacek [8] and Vacek [159]. In [159] the maximum size of such graphs of prescribed radius and order is investigated. It is easy to see that the graphs of maximum size among graphs of given order and radius described after Theorem 4 are radius-edge-invariant. Hence the maximum size of such a graph equals the maximum size of a graph of given order and radius. In [158], Vacek considers graphs with the property that after deletion of any vertex, the resulting graph is radius edge-invariant. Walikar, Buckley and Itagi [162] consider the effect of edge-contraction on the radius. They call an edge whose contraction reduces the radius radius-essential. Among other results they give bounds on the number of radius-essential edges in terms of order and diameter. Fajtlowicz [71] defines a connected graph G to be radius critical if every connected induced proper subgraph of G has radius less than rad(G). He shows that every radius-critical graph is isomorphic to what he calls a ciliate. A ciliate is a graph obtained from p disjoint paths, each on q vertices, where p, q ≥ 1, by choosing an end vertex of each path and linking these together to form a cycle of length q. The question by how much deleting an edge can increase the radius was answered by Segawa [148]. He proved that if a set F of edges is deleted from a graph G such that G−F is connected, then rad(G−F ) ≤ (|F |+1)rad(G)−|F |/2. The effect of removing a vertex or edge, or of adding a new edge, on the radius was considered by Dutton, Medidi and Brigham [61]. 4 Eccentricity

The eccentricity of a vertex v in a connected graph G is deﬁned as the maximum of the distances between v and all other vertices of G. Most research on eccentricities focuses on the eccentric sequence. If G is a connected graph with vertex set {v1, v2, . . . , vn}, and if vertex vi has eccentricity ai, then the sequence a1, a2, . . . , an is the eccentric sequence of G. We always assume that the ai are ordered in nondecreasing sequence, so a1 is the radius of G, and an is the diameter of G. The eccentric sequence is, after the degree sequence, the second oldest sequence associated with the vertices of a graph. It was ﬁrst studied by Lesniak [119]. A sequence is said to be eccentric if it is the eccentric sequence of some graph. Lesniak [119] gave a necessary condition for a sequence of nonnegative integers to be eccentric.

Theorem 9. If a nondecreasing sequence of integers a1, a2, . . . , an is the eccentric sequence of a graph, then

n (i) a1 ≤ 2 ,

(ii) every integer k with a1 < k ≤ an appears at least twice in the sequence a1, . . . , an,

(iii) an ≤ min{n − 1, 2a1}.

Proof. Let G be a graph with eccentric sequence a1, a2, . . . , an. Statement (i) can be re- n stated as rad(G) ≤ 2 , which is Corollary ?? in the section on the radius. Statement (iii) can be restated as diam(G) ≤ min{2rad(G), n − 1}, which clearly holds. To see statement (ii) note that the eccentricities of adjacent vertices diﬀer by not more than 1. Hence a path from a vertex of eccentricity a1 to a vertex of eccentricity an contains vertices of each eccentricity between a1 and an. Let k with a1 < k ≤ an. By the above there exists a vertex u with eccentricity k. Let v be at distance k from u. Then ec(v) ≥ k. Let w be a central vertex, so ec(w) < k. A shortest (v, w)-path hence contains a vertex v0 of eccentricity k. It remains to show that v0 6= u. But

0 this follows from d(v, u) = k and d(v, v ) ≤ d(v, w) = a1 < k. QED

We note that the conditions given above are necessary for a sequence to be eccentric, but not sufficient. Lesniak [119] gave the sequence 2, 3, 3, 3 as an example of a sequence that satisfies the above theorem but is not eccentric. In the same paper she gave the following necessary and sufficient condition for a sequence to be eccentric. For this section we define a subsequence of a sequence S to be either S itself, or a sequence S0 obtained from S by omitting some entries such that S0 has the same number of distinct entries as S. Theorem 10. Let S be a nondecreasing sequence of positive integers. Then S is eccentric if and only if some subsequence of S is eccentric.

Proof. Clearly, if S is eccentric, then an eccentric subsequence of S is the sequence S itself. To show the converse let S be a sequence and let S0 be a subsequence of S which is eccentric. Let G0 be a graph that has S0 as its eccentric sequence. If, say, an entry a appears r times less in S0 than in S, then we choose a vertex v in G0 with eccentricity a, and we add r copies of v to G0, i.e., we add r new vertices that are adjacent to exactly the neighbours of v in G0, and in addition we join all new r vertices to each other and to v. This does not change any eccentricity in G0, but adds r new vertices of eccentricity a. Repeating this step eventually yields a graph with eccentric sequence S. QED As pointed out in [24], Theorem 10 is not always useful if one wishes to determine whether a given sequence is eccentric or not, since the eccentric subsequence may be the original sequence. Hence such sequences are of particular interest. We define an eccentric sequence S to be minimal if there is no proper subsequence of S which is eccentric. Minimal eccentric sequences with two values, i.e., minimal eccentric sequences of the form ah, bk (i.e., a repeated h times and b repeated k times) were considered by Hrnˇciarand Monoszová[98]. They showed that there are exactly seven minimal eccentric sequences of the form 4h, 5k, viz. 47, 52,; 46, 54; 45, 56; 44, 58, 43, 59, 42, 512; and 4, 514. They conjectured that in general there exist 2a − 1 minimal eccentric sequences of the form ah, (a + 1)k. Buckley [22] reports that Nandakumar [134] determined all minimal eccentric sequences with least eccentricity 1 or 2. All 13 minimal eccentric sequences with least eccentricity 3 were determined by Haviar, Hrnˇciarand Monoszová[94]. To date no characterisation is known for minimal eccentric sequences. While no straightforward method is known that determines if for a given sequence S there is a connected graph that has S as an eccentric sequence, the following characterisation due to Lesniak [119] makes it easy to determine if a sequence is an eccentric sequence of a tree.

Theorem 11. A nondecreasing sequence of positive integers a1, a2, . . . , an is the eccentric sequence of a tree if and only if

(i) every number k with a1 < k ≤ an appears at least twice in the sequence a1, . . . , an,

(ii) either a1 = an/2 and a1 < a2, or a1 = a2 = (an + 1)/2 and a1 < a3.

Proof. Necessity in the characterisation follows almost directly from Theorems 9 and 3. To prove suﬃciency, consider a path of length an and note that appending, say, r end vertices adjacent to an internal vertex of the path of eccentricity, say, a, doest not change any eccentricities, but creates r new vertices of eccentricity a + 1. Clearly we can obtain every sequence satisfying (i) and (ii) by repeated appending of end vertices. QED

From the construction sketched above it follows that a sequence that is the eccentric sequence of some tree is also the eccentric sequence of some caterpillar (deﬁned as a graph obtained from a path by appending end vertices to the vertices of the path). Skurnick [149] determined the number of caterpillars that have a given sequence as their eccentric sequence. We mention an inequality by Behzad and Simpson [12], relating the order, size, and the eccentric sequence of a graph G, which follows from deg(v) ≤ n − ec(v) by summation over all v ∈ V (G): n 1 X m(G) ≤ n2 − a . 2 i i=1 A notion related to the eccentric sequence is the eccentric set of a graph G, which was introduced by Behzad and Simpson [12]. It is deﬁned as the set of the eccentricities of the vertices of G. It follows from Theorem 9 that the eccentric set of any connected graph G consists of consecutive integers, the smallest of which is the radius of G, and the largest of which is the diameter of G. The fact that there exists a connected graph with radius r and diameter d if and only if r ≤ d ≤ 2r immediately yields that a set {a1, a2, . . . , ak} (with the ai in increasing order) is the eccentric set of some graph if and only if the set consists of consecutive positive integers and k ≤ a1 + 1, a result given in [12]. Since the eccentric set of a graph G is determined by its radius and diameter, the minimum order of a graph with given eccentric set, as determined in [12], can be directly derived from a result by Ostrand [141] on the minimum order of graphs with given radius and diameter. Mubayi and West [128] considered not the whole eccentric sequence of a graph, but the separate terms. They gave sharp lower bounds and almost sharp upper bounds on the number of vertices of given eccentricity in terms of order and diameter. We mention a result by Nandakumar and Parthasarathy [135] on the eccentricities of vertices in spanning trees. It states that a connected graph G has an eccentricity preserving spanning tree, i.e., a spanning tree T with ecT (v) = ecG(v) for all v ∈ V (G), if an only if each non- central vertex is adjacent in G to a vertex of smaller eccentricity and in addition one of the following two conditions holds: (i) diam(G) = 2rad(G) and G has only one central vertex, or (ii) diam(G) = 2rad(G) − 1 and the centre of G consists of exactly two vertices which are adjacent. There are also some results on graphs that have an eccentricity-approximating spanning trees, i.e., spanning trees in which the eccentricity of no vertex is more than k higher, for some given integer k, than its eccentricity in the original graph. Prisner [145] showed that every chordal graph G has spanning tree with the property that the eccentricity of any vertex in the spanning tree is at most two more than its eccentricity in G. His paper is a good starting point for the literature on similar spanning trees in other graph classes. Two further graph parameters relating to eccentricities in graphs are the average eccentricity [47] and the eccentricity sum. EXPAND THIS, USING THE REFERENCES 1-4 BELOW

5 Wiener index

The Wiener index was the first topological index, and in a way it is the prototype for many distance based topological indices. It has been investigated in the mathematical, chemical and computer science literature under many different names and with slight differences in the definition. The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of vertices, it is sometimes also called the distance or the total distance of a graph. The transmission, status, or routing cost of a graph is the sum of the distances between n all ordered pairs of vertices, hence twice the Wiener index. For the term W (G)/ 2 the names average distance and mean distance are in use. There is an enormous body of literature on the Wiener index, and a comprehensive survey would be beyond the scope of this chapter. We will instead focus on a small number of properties and results that have relevance for other distance based topological indices. The minimum and maximum value of the Wiener index of a graph of given order are attained for the complete graph and the path, respectively. Any two vertices in a connected graph are at distance at least 1, and so we have the lower bound below:

n n + 1 ≤ W (G) ≤ . 2 3

The upper bound on the Wiener index is easily proved by induction. Let G have maximum Wiener index among all graphs of order n. Since deletion of any edge increases the Wiener index, G is a tree and has a vertex v of degree 1. Denote the sum of all distances between v and other vertices by d(v). Then d(v) ≤ 1 + 2 + 3 + ... + (n − 1). By induction we have n W (G − v) ≤ 3 . Hence we obtain

X X W (G) = dG(v, w) + dG(u, w) w∈V −{v} {u,w}⊆V −{v} = d(v) + W (G − v) n ≤ 1 + 2 + ... + (n − 1) + 3 n + 1 = . 3

Clearly, equality in the lower bound holds if and only if every distance equals 1, i.e., if G is complete. Equality in the upper bound implies that d(v) = 1 + 2 + ... + (n − 1), and so G must be a path. This result can be found in [63, 57, 121]. Now consider bounds on the Wiener index of graphs of given order n and size m. Since G n has exactly m pairs of vertices at distance 1, and for the remaining 2 − m pairs the distance is at least 2, we obtain

n W (G) ≥ m + 2 − m = n(n − 1) − m. 2 as observed by Doyle and Graver [57]. Clearly equality holds if an only if diam(G) ≤ 2. As Plesn´ık[144] points out, this bound is very useful as it yields sharp lower bounds on the Wiener index for several graph classes, for example for trees, planar graphs, outerplanar graphs, k- colourable graphs and Kk-free graphs, as pointed out by Plesn´ık[144]. An upper bound in terms of order and size was determined by Soltés[151]. The extremal graph is a path-complete graph, i.e., a graph obtained from the union of a path and a complete graph by joining one end vertex of the path to some vertices of the complete graph, SEE FIGURE ?? A similar bound for bipartite graphs was found by Dankelmann, Dlamini and Swart [41]. The extremal graph in Figure ??? has a shape that is typical for graphs maximising the Wiener index given certain constraints. An example for another typical shape of graphs maximising the Wiener index arises from the conjecture by the computer programmes GRAFFITI n−1 [72, 73] that the average distance of a graph (i.e., W (G) 2 ) is not larger than its independence number. This conjecture, which relates two seemingly unrelated quantities, was proved by Chung [33]. A strengthening of her result is due to Dankelmann [36], who determined the graph maximising the Wiener index among all graphs of given order and independence number. It is obtained from the union of a path and two complete graphs whose cardinality differs by at most one by joining each end of the path to one vertex of one of the complete graphs (see Figure???). For several parameters, such as matching number, domination number [37], generalised packing number [39] the graphs maximising the Wiener index for given order and a prescribed value of this parameter has a similar shape. A third typical shape of graphs that maximise the Wiener index arises in connection with graphs of given order and minimum degree. Since the graphs with largest Wiener index are paths, which have minimum degree 1, it is reasonable to expect that the Wiener index of graphs with larger minimum degree is significantly smaller than that of paths. Kouider and Winkler [113] proved the following bound.

Theorem 12. Let G be a graph of order n and minimum degree δ. Then

n3 W (G) ≤ + O(n2), 2δ + 2 and this bound is best possible.

Graphs of minimum degree δ that show that the bound is best possible are sequential joins of the form Kδ + K1 + K1 + Kδ−1 + +K1 + K1 + Kδ−1 + ... + +K1 + K1 + Kδ−1 + K1 + K1 + Kδ. An example for δ = 4 is shown in Figure ???. This somewhat pathlike shape of this graph is the third typical shape of extremal graphs maximising the Wiener index under certain conditions. We note the above Theorem was strengthened by Dankelmann and Entringer [43] in the sense

n3 2 that it was shown that G even has a spanning tree T with W (T ) ≤ 2δ+2 +O(n ). Stronger bounds for triangle-free graphs and C4-free graphs were also given in [43]. Similar bounds for K3,3-free graphs and K2,t-free graphs were proved by Dankelmann, Swart and Dlamini [41, 42, 40]. n−1 We comment brieﬂy on the relationship between the average distance (deﬁned as 2 W (G), which is more appropriate than the Wiener index for this question) and the two oldest distance parameters: diameter and radius. Since every distance is at most as large as the diameter, we have for the average distance, the following inequality:

1 ≤ µ(G) ≤ diam(G).

The question if this inequality can be improved was answered in the negative by Plesn´ık[144]. He constructed graph with prescribed radius and diameter, so that for every given real t and every small positive real there exists a graph whose average distance is within of t. Finally we consider the Wiener index of spanning trees. Clearly, if G is not a tree, the Wiener index of any spanning tree of G is greater than the Wiener index of G. However, Entringer, Kleitman and Sz´ekely [65] showed that it is always possible to ﬁnd a spanning tree whose Wiener index is not much larger than that of G.

Theorem 13. Let G be a connected graph. Then there exists a spanning tree of G such that W (T ) ≤ 2W (G).

The proof is a simple averaging argument. For each vertex v of G let Tv be a distance preserving spanning tree of G, rooted at v. Then

X W (Tv) = dTv (x, y) {x,y}⊆V X ≤ dTv (x, v) + dTv (v, y) {x,y}⊆V X ≤ dG(x, v) + dG(v, y) {x,y}⊆V X = (n − 1) dG(v, w). w∈V Summing over all vertices v of G we obtain that

X X X n − 1 W (T ) ≤ (n − 1) d (v, w) = W (G), v G 2 v∈V v∈V w∈V

n−1 and so there exists a vertex v for which W (Tv) ≤ 2n W (G). QED

For some applications we wish to ﬁnd a spanning tree of minimum average distance of a given input graph G, a so-called MAD tree. The problem of ﬁnding a MAD tree of a given graph has been proved to be NP-complete [107]. However, the MAD tree problem can be solved in polynomial time for some graph classes (see [34, 35]). There is also a polynomial time approximation scheme [164] for this problem.

6 Eccentric connectivity index

An important tool in pharmaceutical drug design is the prediction of physico-chemical, phar- macological and toxicological properties of a compound directly from its molecular structure. This information is used to select the most promising compounds for a desired property, and hence decrease the number of compounds which need to be synthesized during the process of designing new drugs [7, 84, 117]. Many topological indices have been defined, used and have shown to give a high degree of predictability of pharmaceutical properties, and so may provide leads for the development of safe and potent drugs, such as anti-HIV compounds. The first topological index, the Wiener index, and most well-known parameter, was introduced in the late 1940’s in an attempt to analyze the chemical properties of paraffins (alkanes) [163]. The Hosoya index, Randić’smolecular connectivity index, Zagreb group parameters and Balaban’s index were introduced in the 1970’s and 1980’s [60]. Dozens of other topological descriptors can be found in the literature. In this section, we will discuss one of the newest distance-based topological index, the eccentric connectivity index, conceptualized in 1997 by Sharma, Goswami and Madan [139]. Experi- ments reveal that prediction, using eccentric connectivity index, of analgesic activity [139] and of anti-inflammatory activity [86] is more superior than that predicted by the Wiener index.

Deﬁnition 1. Let G be a connected graph with vertex set V . The eccentric connectivity

C P index ξ (G) of G is deﬁned as v∈V deg(v) ec(v) where deg(v) is the degree of vertex v and ec(v) is its eccentricity.

Mathematical properties of the eccentric connectivity index is presently a topic of intense study [6, 5, 56, 103, 124, 125, 126, 138, 169]. Exact formulas for the eccentric connectivity index for graphs of chemical interest, such as the TUC4C8(S) nanotube and TC4C8(S) nanotorus constructed from squares and octagons by leapfrog operation (see, for example [6]), and the

TUC4C8(R) nanotubes [5], have also been computed. Where exact formulae cannot be obtained, extremal values of the molecular descriptors is of paramount importance and is closely linked to isomer enumeration [120]. Assume that an integral index X is shown to have minimum and maximum values of Xm and XM respectively, and that a particular class of chemical compounds under consideration has N isomers. If N > (XM − Xm) , then two or more isomers will have the same value of the chosen index X. This type of ‘degeneracy’ is a serious problem encountered with topological indices. The eccentric connectivity index has been found to have quite low degeneracy [56]. This fact, along with simplicity of required computations, makes it potentially very useful for predicting various properties of many classes of chemical compounds. We will explore most of the known mathematical results to date and point out on some of the open problems. We begin by presenting some graphs extremal with respect to the eccentric connectivity index. 6.1 Some extremal graphs

Formulas for the eccentric connectivity index of the complete graph Kn, complete bipartite graph Kp,q, cycle graph Cn, star graph Sn and the path Pn have been calculated independently by several authors [56, 124, 169].

Deﬁnition 2. The broom graph Bn,d consists of a path Pd , together with n − d end vertices all adjacent to the same end vertex of Pd .

Deﬁnition 3. The lollipop graph Ln,d , also known as the kite graph, is obtained from a complete graph Kn−d and a path Pd , by joining one of the end vertices of Pd to all the vertices of Kn−d.

Deﬁnition 4. The volcano graph Vn, d is a graph obtained from a path Pd+1 and a set S of n − d − 1 vertices, by joining each vertex in S to a central vertex of Pd+1.

V11,6

B9,4 L9,4

Figure 1: Graphs B9,4, L9,4, V11,6.

Straightforward calculations show that

 2dn − n − d2/2 − d + 1 for d even C  ξ (Bn,d) =  1 2 2 (3 − 2d − d − 2n + 4dn) for d odd;

 1 2 3 2 2 2 (2 − 2d + d + 2d − 2n + 2dn − 4d n + 2dn ) for d even C  ξ (Ln,d) =  1 2 3 2 2 2 (3 − 2d + d + 2d − 2n + 2dn − 4d n + 2dn ) for d odd;  nd + n + d2/2 − 2d − 1 for d even C  ξ (Vn,d) =  nd + 2n + d2/2 − 3d − 3/2 for d odd.

6.2 Bounds in terms of order

It was proved independently by Morgan, Mukwembi and Swart [124], as well as Zhou and Du [169], that the minimum eccentric connectivity index for a graph of order n is attained by the star graph. The proof is based on the simple observation that end vertices cannot be centre vertices and vertices of maximum degree n − 2, where n is the order of the graph, have eccentricity at least 2. Despite being a simple proof technique, several authors have developed it further and accomplished more complicated bounds on the eccentric connectivity index.

Theorem 14. [124, 169] Let G be a connected graph of order n, n ≥ 4. Then

ξC (G) ≥ 3(n − 1) with equality if and only if G is the star graph.

Proof. Partition the vertices of G into three sets: A = {v ∈ V | deg(v) = n − 1}, B = {v ∈ V | n − 2 ≥ deg(v) ≥ 2} and C = {v ∈ V | deg(v) = 1}. Then letting |A| = a, |B| = b and |C| = c, we obtain

a + b + c = n. (1)

Since deg(v) ≤ n − 2 for every vertex v in B ∪ C, it is easy to see that, for n ≥ 4,

ec(v) ≥ 2 for all v ∈ B ∪ C. (2)

If on one hand, A 6= ∅, i.e., a ≥ 1, then (1) and (2) in conjunction with n > 3, give

X X ξC (G) = ec(v) deg(v) + ec(v) deg(v) v∈A v∈B∪C X X ≥ 1 · (n − 1) + 2 · 1 v∈A v∈B∪C = a(n − 1) + 2(b + c)

= 2n + a(n − 3)

≥ 2n + n − 3, as claimed.

If on the other hand A = ∅, i.e., a = 0, then it can be seen that ec(v) ≥ 3 for all v ∈ C. This, together with (1) and (2) yields

X X ξC (G) = ec(v) deg(v) + ec(v) deg(v) v∈B v∈C X X ≥ 2 · 2 + 3 · 1 v∈B v∈C = 4b + 3c

= 3n + b, and the bound is established. 2

Turning to upper bounds, an asymptotic maximum of the eccentric connectivity index was independently determined by Morgan, Mukwembi and Swart [124], and by Doˇsli´c,Saheli & Vukiˇcevi´c[56]. They proved that the eccentric connectivity index grows no faster than the cubic polynomial in the number of vertices, and in addition, constructed a family of graphs for which the leading term of the cubic bound is attained.

Theorem 15. [124, 56] Let G be a connected graph of order n. Then

4 ξC (G) ≤ n3 + O(n2), 27 with equality if G is a lollipop graph L n . n, 3

For trees, the above upper bound can be improved. Not surprisingly, the maximum index value amongst trees of order n is attained by the path graph Pn [56, 124, 169].

6.3 Bounds in terms of order and diameter

The diameter, as well as the radius, has been used as an important tool when dealing with topological indices. The diameter, when prescribed, has aided in the determination of bounds on the eccentric connectivity index. The starting point in this direction is a very simple result on trees reported independently by Morgan, Mukwembi and Swart [124], and Zhou and Du [169].

Theorem 16. [124, 169] Let T be a tree of order n and diameter d. Then

C C ξ (T ) ≥ ξ (Vn,d).

In [125], this lower bound was proved to be true for all graphs. They proved the general result that if G is a connected graph of order n and diameter d, then

C C ξ (G) ≥ ξ (Vn,d).

This simple generalization is quite challenging to prove; the diﬃculty being with some ‘problem vertices’, i.e., vertices of degree 2 having eccentricity dd/2e. For some graphs there may be an enormous number of such problem vertices and more harder will be to deal with their contribution to the eccentric connectivity index. The proof presented in [125] is long, involving and highly technical. We therefore refer the reader to [125]. Turning to upper bounds on the eccentric connectivity index of G in terms of order n and diameter d, let P be a diametral path of G and let S be the set of n−d−1 vertices in G that are not on P . It is not hard to verify (see, for example [124]) that the total contribution of vertices on P to the eccentric connectivity index of G is at most O(n2). Evidently, every vertex in S cannot be adjacent to more than three vertices on P and so has degree at most n − d + 1. It follows that the contribution of vertices in S to the eccentric connectivity index of G is at most |S|(n − d + 1)d = d(n − d)2 + O(n2), and this yields the following upper bound proved in [124].

Theorem 17. [124] Let G be a connected graph of order n and diameter d. Then

ξC (G) ≤ d(n − d)2 + O(n2),

with equality if G is a lollipop graph Ln,d.

This bound, by a simple maximization of the term d(n − d)2 + O(n2) with respect to d, can be used to deduce the bound presented in Theorem 15. For trees, Theorem 17 can be improved.

Theorem 18. [124] Let T be a tree of order n and diameter d. Then

C C ξ (T ) ≤ ξ (Bn,d).

6.4 Bounds in terms of order and size

Evidently, if G is a connected graph of size m, radius r and diameter d, then

2mr ≤ ξC (G) ≤ 2md.

The lower bound was improved by Zhou and Du [169] who proved that if G is a connected graph n of order n and size m, n − 1 ≤ m < 2 , then $ % 2n − 1 − p(2n − 1)2 − 8m ξC (G) ≥ 4m − (n − 1) , 2 and that the bound is best possible. They also posed the question of determining an upper bound on the eccentric connectivity index in terms of order and size. This question is quite elusive, and below we present a ﬁrst attempt to address the problem.

Theorem 19. Let G be a connected graph of order n and size m. Then

4m2 ξC (G) ≤ 2nm − . n Proof. Note that for each vertex x, ecG(x) ≤ n − degG(x). Thus

C X ξ (G) = ecG(x)degG(x) x∈V X ≤ (n − degG(x))degG(x) x∈V X X 2 = n degG(x) − [degG(x)] x∈V x∈V X 2 = 2nm − [degG(x)] . x∈V By the Cauchy-Schwarz inequality,

P 2 2 X [ degG(x)] [2m] [deg (x)]2 ≥ x∈V = , G n n x∈V and the upper bound follows. 2

6.5 Regular graphs

A k-regular graph is a graph in which every vertex has degree k. Very recently, Doˇslić,Saheli & Vukiˇcević[56] remarked that it would be interesting to determine extremal cubic graphs, cubic graphs being 3-regular graphs, with respect to the eccentric connectivity index; and more generally, Ilić[103] has posed the same question for all regular graphs with special emphasis on cubic graphs. This problem is addressed in [126] by Morgan, Mukwembi and Swart. They determined, completely, a tight upper bound on the eccentric connectivity index of regular graphs of given order and constructed extremal graphs. The derivation of the upper bound draws from Dankelmann, Goddard and Swart [47]’s bound

9n 15 avec(G) ≤ + , 4(δ + 1) 4 on the total eccentricity avec(G) of a graph G in terms of order n and minimum degree δ. Eﬀortlessly, from this bound one deduces

Theorem 20. [126] Let G be a k-regular connected graph of order n. Then, for k ≥ 3,

9 k n2 ξC (G) ≤ + O(n) 4(k + 1) and the bound is sharp.

A lower bound is rather tricky to obtain. A starting point to addressing this problem was presented in [126]. Theorem 21. [126] Let G be a k-regular connected graph of order n. Then, for k ≥ 2,

C ξ (G) ≥ kn[logk−1(n(k − 2) + 2) − logk−1 k].

For cubic graphs G of order n, this bound reduces to

C ξ (G) ≥ 3n[log2(n + 2) − log2 3], (4) which is attained by the Petersen graph, a graph of diameter 2. For d ≥ 3, this bound, (4), is not best possible, and the problem of improving it joins several other open degree-diameter ﬂavoured problems in literature.

7 Degree distance

The degree distance, a variant of the Wiener index, is deﬁned by

X D0(G) = (deg(u)deg(v))d(u, v). {u,v}⊂V

Its study was proposed by Dobrynin and Kochetova [55] and independently, under a different name, by Gutman [87]. It appears that this quantity had also been mentioned in connection with certain chemical applications. The degree distance has recently been investigated in more detail in the mathematical literature. The degree distance can be considered a weighted version of the Wiener index. For trees of given order, these two parameters actually determine each other: Klein, Mihalić,Plavˇsić, Trinajstić[112] and also Gutman [87] showed that, for every tree T of order n,

D0(T ) = 4W (T ) − n(n − 1). (5)

We give a proof which uses a technique applicable to many similar situations: Fix an edge uw of T and compare the contribution of this edge to the Wiener index and to the degree distance. Let U and W be the vertex sets of the components of T − uw containing u and w, respectively, and let nu,uw and nw,uw be their respective cardinalities. The Wiener index is the total length n of shortest paths between the 2 unordered pairs of vertices. Clearly, nu,uwnw,uw of these paths P contain uw, and so W (T ) = uw∈E(T ) nu,uwnw,uw. The degree distance can be seen as the total lengths of shortest paths between all unordered pairs of vertices, where we have deg(x) + deg(y) shortest paths between x and y. The number of such paths containing uw is then

X X X (deg(x) + deg(y)) = |W | deg(x) + |U| deg(y) x∈U,y∈W x∈U y∈W = |W |(2|U| − 1) + |U|(2|W | − 1)

= 4nu,uwnw,uw − n.

Summation over all edges now yields (5). As a consequence of (5) and the fact that the Wiener index is maximised for paths, and minimised for stars, we obtain sharp lower and upper bounds on the degree-distance of a tree T of order n: n(n − 1)(2n − 1) 3n2 − 7n + 4 ≤ D0(T ) ≤ , 3 with equality in the ﬁrst or second inequality if and only if T is a star K1,n−1 or a path Pn, respectively (see [154]). While it is straightforward to determine the trees of given order maximising/minimising the degree distance, this is by no means true for general graphs. Dobrynin and Kochetova [55]

n4 3 conjectured that the largest degree distance of all connected graphs of order n equals 32 +O(n ). This was refuted by Tomsecu [154], who showed that graphs consisting of two cliques of order

n4 approximately n/3 joined by a path on approximately n/3 vertices, have degree distance 27 + O(n3). He conjectured this to be the correct value of the maximum degree distance, but was

0 2n4 3 able to prove only the upper bound D (G) ≤ 27 + O(n ). Tomescu’s conjecture was proved, except for a slightly weaker error term, by Dankelmann, Gutman, Mukwembi and Swart [49] who gave an upper bound on the degree distance in terms of order and diameter, and used this

0 n4 7/2 to show that D (G) ≤ 27 +O(n ). The smallest value of the degree distance of graphs of order n is 3n2 − 7n + 4, attained by stars, as conjectured in [55] and proved by Tomescu [154] (see also the next paragraph). In another paper [156], Tomescu showed that the second and third smallest degree distance among graphs of order n is attained by the graph obtained from the star K1,n−2 by subdividing an edge, and the graph K1,n−1 + e, respectively. We now determine the minimum degree distance among connected graphs of given order n and size m. For a vertex v of a connected graph G there are exactly deg(v) vertices at distance 1, and the remaining n − 1 − deg(v) vertices have distance at least 2 from v. Hence

X deg(v) d(v, x) ≥ deg(v)(deg(v) + 2(n − 1 − deg(v))) = deg(v)(2n − 2 − deg(v)). x∈V (G) If the degrees of the vertices of G are d1, d2, . . . , dn, then we obtain

n 0 X D (G) ≥ di(2n − 2 − di). (6) i=1 We ﬁrst use (6) to determine the minimal value of D0(G) over all graphs of order n. Clearly all Pn 1 ≤ di ≤ n−1 for all i, and i=1 di ≥ 2n−2. Deﬁne a real function f(x) by f(x) = x(2n−2−x). Since f(x) is increasing and concave for 1 ≤ x ≤ n − 1, it follows from Jensen’s inequality that Pn Pn i=1 f(xi) is minimised, subject to i=1 xi ≥ 2n − 2 and 1 ≤ xi ≤ n − 1 for i = 1, 2, . . . , n − 1, if one of the values xi equals n − 1, while the remaining xj for j 6= i equal 1. Hence

D0(G) ≥ (n − 1)(n − 1) + (n − 1)(2n − 3) = 3n2 − 7n + 4,

with equality holding if and only if G is a star K1,n−1. This result is due to Tomescu [154]. Inequality (6) can also be used to obtain the minimum degree distance over all graphs of given order and size. Rewriting (6) we obtain

X D0(G) = (2n − 2)2m − deg(v)2. v∈V (G)

P 2 The quantity v∈V (G) deg(v) has featured repeatedly in the graph theory literature, usually simply called the sum of the squares of the degrees, but also the name ﬁrst Zagreb index has been used. Ahlswede and Katona [1] (see also [14] and [142]) determined the extremal graphs maximising the sum of the squares of the degrees. Bucicovschi and Cioab´a[21] pointed out that these extremal graphs have a vertex of degree n − 1 and hence have diameter at most 2. They thus derived the sharp bound

D0(G) ≥ (2n − 2)2m − g(n, m), (7) where g(n, m) is the maximum of the sum of the squares of the degrees over all graphs of order n and size m. An equivalent result was obtained by Tomescu [155]. Using (7) and the values g(n, n) = n2 − n + 6 and g(n, n + 1) = n2 − n + 4, we obtain the values of the minimum degree distance over all

2 (i) unicyclic graphs: 3n − 3n + 6, and the unique extremal graph is the graph K1,n−1 + e (see [153]),

2 (ii) bicyclic graphs: 3n + n − 18, and the unique extremal graph is obtained from a star K1,n−1 by adding two incident edges (see [153]). The value of g(n, m) is not a simple expression, so we replace it by a (for m >> n) good,

P 2 2m2 but not sharp upper bound due to de Caen [51], v∈V (G) deg(v) ≤ n−1 + m(n − 2), to obtain the general bound 2m2 D0(G) ≤ (3n − 2)m − . n − 1 The determination of the maximum degree distance among graphs of given order n and size m seems a diﬃcult problem. So far it has been solved only for two special cases, unicyclic graphs (see Hou and Chang [97]) and bicyclic graphs (see Iliˇc,Stevanoviˇc,Feng, Yu and Dankelmann [104]).

Theorem 22. Let G be a connected graph of order n and size m.

0 0 (i) If m = n then D (G) ≤ D (Hn), where Hn is the graph of order n obtained from a triangle

K3 and a path Pn−2 by identifying an end vertex of the path with a vertex of the triangle.

0 0 0 0 (ii) if m = n + 1 then D (G) ≤ D (Hn), where Hn is the graph of order n obtained from two disjoint triangles K3 and a path Pn−4 by identifying each end vertex of the path with a vertex of one of the two triangles. The extremal graphs in (i) and (ii) are unique.

We note that above result for unicyclic graphs was reﬁned by Du and Zhou [58] who gave a sharp upper bound on the degree distance of unicyclic graphs of given order and maximum degree. The degree distance of unicyclic graphs of given order and girth was considered in [104]. Further results on the degree distance are given in [96].

8 Gutman index

Several variants of the Wiener index have been proposed and studied. In this section we consider a variant of the Wiener index, a quantity put forward in [87] by Gutman and called there the Schultz index of the second kind, but for which the name Gutman index has also sometimes been used [152].

Definition 5. Let G be a connected graph with vertex set V . The Gutman index Gut(G) is defined as X Gut(G) := deg(x) deg(y) d(x, y). {x,y}⊆V The Gutman index is thus a kind of a vertex-valency-weighted sum of the distances between pairs of vertices in a graph, and as shown below, for acyclic structures the index is closely related to the Wiener index and reflects precisely the same structural features of a molecular as the Wiener index does.

Theorem 23. [87] Let T be a tree of order n. Then

Gut(T ) = 8W (T ) − 2(2n − 1)(n − 1).

Gutman [87] remarked that theoretical investigations on the Gutman index focusing on polycyclic molecules is more diﬃcult. Since then, several authors [4, 32, 76, 87, 100] have studied the index and also its relationship with other graph parameters. One quantity closely analogous to the Wiener index which is connected to the Gutman index is the edge-Wiener index

We(G) deﬁned in [48] as the sum of the distances (in the line graph) between all pairs of edges of G. The Gutman index is connected to this quantity by the following inequality.

Theorem 24. [48] Let G be a connected graph of order n. Then

1 n4 W (G) − Gut(G) ≤ . e 4 8

Bounds on the Gutman index in terms of order have also been researched on. Expectedly, the star graph has the smallest Gutman index.

Theorem 25. [4] Let G be a connected graph of order n. Then

Gut(G) ≥ (2n − 3)(n − 1)), with equality if and only if G is the star graph.

Upper bounds are more complicated. Recently, Dankelmann, Gutman, Mukwembi and Swart [48] presented an upper bound on the Gutman index of a graph in terms of its order. It was shown

n that the graph Gn of order n, where n is a multiple of 5, obtained from a path with 5 vertices 2n and two vertex disjoint cliques of order 5 by adding two edges, each joining an end vertex of the 24 5 4 path to a vertex in a clique, is extremal with respect to the index and Gut(Gn) = 55 n + O(n ).

Theorem 26. [48] Let G be a connected graph of order n. Then

24 Gut(G) ≤ n5 + O(n9/2), 55 and the coeﬃcient of n5 is best possible. One would expect that the O(n9/2) in the bound can be replaced by an O(n4) term to

24 5 4 9/2 obtain an improved tight bound, Gut(G) ≤ 55 n + O(n ). Replacing the O(n ) term from the bound by an O(n4) term turns out to be quite challenging. Recently, Mukwembi [129] carefully engineered a counting technique which can adequately capture the contribution to the index made by pairs of vertices with at least one vertex on a diametral path and proved indeed that

24 Gut(G) ≤ n5 + O(n4). 55

For trees this bound can be improved.

Theorem 27. [4] Let T be a tree of order n. Then

(n − 1)(2n2 − 4n + 3) Gut(T ) ≤ , 3 and equality is attained by the path.

Bounds on the Gutman index for graphs with minimal or maximal Gutman index in terms of order, minimum degree and maximum degree were presented in [4].

9 Eccentric Distance Sum

The eccentric distance sum, a novel topological index that oﬀers a vast potential for structure activity/property relationships, was conceptualized by Gupta, Singh and Madan [85]. There it was uncovered, in the structure-activity and quantitative structure-property studies, that predictions using eccentric distance sum were better than using Wiener index. Formally,

Deﬁnition 6. The eccentric distance sum ξd(G) of a connected graph G is deﬁned as ξd(G) = P x∈V (G) ecG(x)D(x), where D(x) is the sum of all distances from vertex x and ecG(x) is the eccentricity of x in G.

The investigation of the mathematical properties of eccentric distance sum started only recently [105, 167, 168, 99]. First observe that removing an edge from a graph does not decrease distances. It follows that, as in the case of the Wiener index, for every spanning tree T of G, ξd(G) ≤ ξd(T ). Thus the problem of finding an upper bound on the eccentric distance sum of a graph can now be rectricted to finding an upper bound on the eccentric distance sum of trees. Ilić,Yu and Feng [105] put forward a clever graph transformation which increases the eccentric distance sum. They applied the transformation inductively to trees and proved that d d if G is a connected graph of order n, then ξ (G) ≤ ξ (Pn). The operation is as follows: Let w be a vertex of a nontrivial connected graph G. For nonnegative integers p and q, let G(p, q) denote the graph obtained from G by attaching to vertex w pendant paths P = wv1v2 . . . vp and

Q = wu1u2 . . . uq of lengths p and q, respectively. Let G(p + q, 0) = G(p, q) − wu1 + vpu1. Let r be the eccentricity of the vertex w in G. If r ≥ p ≥ q ≥ 1, then ξd(G(p, q)) ≤ ξd(G(p + q, 0)).

d d Interestingly, the same result, i.e., ξ (G) ≤ ξ (Pn) was independently rediscovered by Zhang and Li [168] who used a diﬀerent graph transformation. They also improved this upper bound for

d d unicyclic graphs and showed that if G is a unicyclic graph of order n > 8, then ξ (G) ≤ ξ (Un), where Un is the graph obtained from the path Pn−1 = v0v1 . . . vn−3vn−2 by joining vertex vn−1 to vn−3 and vn−2. Turning to lower bounds, clearly the complete graph minimizes the eccentric distance sum amongst all graphs of given order. This simple bound on the index was improved for trees and for unicyclic graphs, and in the former the star graph was identiﬁed to be the unique graph which minimizes the index.

Theorem 28. [167] (a) Let T be a tree of order n. Then

ξd(T ) ≥ 4n2 − 9n + 5.

(b) Let G be a unicyclic graph of order n > 5. Then

ξd(G) ≥ 4n2 − 9n + 1, with equality if and only if G is the graph obtained by adding an edge between two end vertices of the star graph of order n.

A natural generalization of trees and unicyclic graphs is the class of the so called cacti graphs, which we will deﬁne shortly. By treating the lower bound for the eccentric distance sum of the cacti graph, Hua, Xu and Wen [99] discovered a short and uniﬁed proof to Theorem 28.

Deﬁnition 7. A cactus graph is a connected graph, each of whose blocks is either a cycle or an edge.

Hua, Xu and Wen’s proof of a lower bound on the eccentric distance sum of a cactus of order n with k cycles is based on a simple observation that a cacti has at most one vertex of eccentricity 1 since it has no cycles sharing common edges. Now if on one hand G has one ∼ vertex of eccentricity 1, then one can show eﬀortlessly that G = Catn,k, where Catn,k is the cactus obtained by introducing k independent edges among end vertices of the star graph of order n. An easy calculation, for this case, reveals that ξd(G) ≥ 4n2 − 9n − 4k + 5. If on the other hand all vertices of G have eccentricity at least 2, then

X X ξd(G) ≥ 2 D(v) ≥ 2[2n(n − 1) − deg(v)] = 4n(n − 1) − 4m, v∈V (G) v∈V (G) and Hua, Xu and Wen’s lower bound, which we state below, now follows from the fact that in a cactus with n vertices and k cycles, there are n + k − 1 edges.

Theorem 29. [99] Let G be a cactus of order n ≥ 4 and with k ≥ 0 cycles. Then

ξd(G) ≥ 4n2 − 9n − 4k + 5,

∼ with equality if and only if G = Catn,k.

Theorem 28 is then rediscovered by setting k = 0 and 1. More interesting is to derive bounds on the eccentric distance sum in terms of order and diameter. As in the case of the eccentric connectivity index, Yu, Feng and Ili´c[167] showed that for trees of order n and diameter d the volcano graph has the smallest eccentric distance sum. Precisely,

Theorem 30. [167] Let T be a tree of order n and diameter d. Then

d d ξ (T ) ≥ ξ (Vn,d).

An analogue of this result for the eccentric connectivity index was generalized for all graphs

d d in [126]. It would be interesting to know whether or not ξ (G) ≥ ξ (Vn,d), for a graph G of order n and diameter d and in addition to establish a sharp upper bound on ξd(G) in terms of the order and diameter of G. Various explicit formulae for the eccentric distance sum of graphs of chemical interest, such as the C4 nanotorus, i.e., a Cartesian product of two cycles, the rectangular grid, the C4 nanotube, i.e., a Cartesian product of the path and the cycle, and for the join of two graphs, were presented in [105]. Further, they established bounds on the eccentric distance sum in terms of other graph invariants, such as the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number and the clique number. We mention here that the mathematical properties of several derivatives of the eccentric distance sum, such as the adjacent eccentric distance sum index deﬁned as ξsv(G) = P ec(v)·D(v) v∈V (G) deg(v) , are yet to be researched on.

References

[1] R. Ahlswede and G.O.H. Katona, Graphs with maximum number of adjacent pairs of edges, Acta Math. Acad. Sci. Hungar. 32 (1978) 97-120.

[2] P. Ali, P. Dankelmann, S. Mukwembi, The radius of 3- and 4-connected planar graphs with bounded faces, (submitted).

[3] D. Amar, I. Fournier, A. Germa, Ordre minimum d’un graphe simple de diametre, degré minimum et connexitédonnés, Ann. Discrete Math. 17 (1983) 7-10.

[4] V. Andova, D. Dimitrov, J. Fink, R. Skrekovski, Bounds on Gutman index, IMFM 49 (2011) (pre-print).

[5] A. R. Ashraﬁ, T. Doˇsli´c,M. Saheli, The eccentric connectivity index of TUC4C8 (R) nanotubes, MATCH Commun. Math. Comput. Chem. 65 (1) (2011) 221-230.

[6] A. R. Ashraﬁ, M. Saheli, M. Ghorbani, The eccentric connectivity index of nanotubes and nanotori, J. Comput. Appl. Math. 235 (16) (2011) 4561-4566.

[7] S. Bajaj, S. S. Sambi, A. K. Madan, Topological models for prediction of anti-HIV activity of acylthiocarbamates, Bioorg. Med. Chem. 13 (2005) 3263-3268.

[8] V. B´alint, O. Vacek, Radius-invariant graphs, Math. Bohem. 129 (2004) 361-377.

[9] E. Bannai, T. Ito, Regular graphs with excess one, Discrete Math. 37 (1981) 147-158.

[10] S. C. Basak, A. T. Balaban, G. D. Grunwald, B. D. Gute, Topological Indices: Their Nature and Mutual Relatedness, J. Chem. Inf. Comput. Sci. 40 (4) (2000) 891-898.

[11] S. C. Basak, S. Bertelsen, G. D. Grunwald, Use of graph theoretic parameters in risk assessment of chemicals, Toxicol. Lett. 79 (1995) 239-250.

[12] M. Behzad, J.E. Simpson, Eccentric sequences and eccentric sets in graphs, Discrete Math. 16 (1976) 187-195. [13] A. Boals, N.A. Sherwani, H.H. Ali, Vertex diameter crtical graphs, Congr. Numer. 72 (1990) 187-192.

[14] F. Boesch, R. Brigham, S. Burr, R. Dutton, R. Tindell, Maximising the sum of the squares of the degrees of a graph, Tech. Rep. Stevens Inst. Tech., Hoboken, New Jersey, 1990.

[15] B. Bollob´as,Graphs with a given diameter and maximal valency and with a minimal number of edges (D. Welsh, ed.), Academic Press (London and New York 1971), pp 25-37.

[16] B. Bollob´as,F. Harary, Extremal graphs with given diameter and connectivity, Ars Combin. 1 (1976) 281-296.

[17] B. Bollob´as,Graphs with given diameter and minimal degree, Ars Combin. 2 (1976) 3-9.

[18] B. Bollob´as,Extremal Graph Theory. London Mathematical Society Monographs 11, Aca- demic Press, London-New York, 1978.

[19] A. Bondy, U.S.R. Murty, Extremal graphs of diameter 2 with prescribed minimum degree, Studia math. Sci. Hungar. 7 (1972) 239-241.

[20] J. Bos´ak,A. Rosa, S. Sn´am,Onˇ decomposition of complete graphs into factors of given diameters, Theory of Graphs (Proc. Colloq. Tihany), Academic Press, New York 1968, 37-56.

[21] O. Bucicovschi, S.M. Cioab´a,The minimum degree distance of graphs of given order and size, Discrete Appl. Math. 156 (2008) 3518-3521.

[22] F. Buckley, Eccentric sequences, eccentric sets, and graph centrality, Graph Theory Notes N. Y. 40 (2001) 18-22.

[23] F. Buckley, M. Lewinter, A note on graphs with diameter-preserving spanning trees, J. Graph Theory 12 (4) (1988) 525-528.

[24] F. Buckley, F. Harary, Distance in Graphs. Addison-Wesley, Redwood City, California, USA, 1989.

[25] L. Caccetta, R. H¨aggkvist,On diameter critical graphs, Discrete Math. 28 (1979) 223-229.

[26] L. Caccetta, W.F. Smyth, Properties of K-edge-connected D-critical graphs, J. Combin. Math. Combin. Comput. 2 (1987) 111-131. [27] L. Caccetta, W.F. Smyth, K-edge-connected D-critical graphs of minimum order, Congr. Numer. 58 (1987) 225-232.

[28] L. Caccetta, W.F. Smyth, Redistribution of vertices for maximum K-edge-connected D- critical graphs, Ars Combin. 26B (1988) 115-132.

[29] L. Caccetta, S. El-Batanouny, On the existence of vertex critical regular graphs of given diameter, Australas. J. Combin. 20 (1999) 145-161.

[30] E. Canale, J. G´omez,Asymptotically large (∆,D)-graphs, Discrete Appl. Math. 152 (1-3) (2005) 89-108.

[31] G. Chartrand, M. Schultz, S. Winters, On eccentric vertices in graphs, Networks 28(4) (1996) 181-186.

[32] S. B. Chen, W. J. Liu, Extremal modiﬁed Schultz index of bicyclic graphs, MATCH Com- mun. Math. Comput. Chem. 64 (2010) 767–782.

[33] F.R.K. Chung, The average distance and the independence number, J. Graph Theory 12 (1988) 229–235.

[34] E. Dahlhaus, P. Dankelmann, W.D. Goddard, H.C. Swart, MAD trees and distance hereditary graphs, Discrete Appl. Math 131 (2003) 151-167.

[35] E. Dahlhaus, P. Dankelmann, R. Ravi, The eﬃcient computation of a minimum average distance spanning tree of an interval graph, Inform. Process. Lett. (to appear)

[36] P. Dankelmann, Average distance and independence number, Discrete Appl. Math. 51 (1994) 75–83.

[37] P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21–35.

[38] P. Dankelmann, A note on MAD spanning trees, J. Combin. Math. Combin. Comput. 32 (2000) 93-95.

[39] P. Dankelmann, Average distance and generalised packing number, Discrete Math. 31(17- 18) (2009) 2334-2344.

[40] P. Dankelmann, G. Dlamini, H.C. Swart, Distances in bipartite graphs. (submitted) [41] P. Dankelmann, G. Dlamini, H.C. Swart, Upper bounds on distance measures in K3,3-free graphs, Util. Math. 67 (2005) 205-221.

[42] P. Dankelmann, G. Dlamini, H.C. Swart, Upper bounds on distance measures in K2,t-free graphs, (submitted).

[43] P. Dankelmann, R.C. Entringer, Average distance, minimum degree and spanning trees, J. Graph Theory 33 (2000) 1–13.

[44] P. Dankelmann, S. Mukwembi, H.C. Swart, An upper bound on the radius of a 3-edge- connected graph, Util. Math. 73 (2007) 207-215.

[45] P. Dankelmann, H.C. Swart, P. van den Berg, Diameter and inverse degree, Discrete Math. 308 (2008) 670–673.

[46] P. Dankelmann, H.C. Swart, P. van den Berg, The number of edges in a biparite graph of given radius, Discrete Math. 311 (2011) 690-698.

[47] P. Dankelmann, W. Goddard, C.S. Swart, The average eccentricity of a graph and its subgraphs, Util. Math. 65 (2004) 41-51.

[48] P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, The edge-Wiener index of a graph, Discrete Math. 309 (2009) 3452–3457.

[49] P. Dankelmann, I. Gutman, S. Mukwembi, H.C. Swart, On the degree distance of a graph, Discrete Appl. Math. 157 (2009) 2773-2777.

[50] P. Dankelmann, L. Volkmann, Minimum size of a graph or digraph of given radius, Inform. Process. Lett. 109(16) (2009) 971-973.

[51] D. de Caen, An upper bound on the sum of squares of degrees in a graph, Discrete Math. 185 (1998) 245-248.

[52] E. Delavi˜na,B. Waller, Independence, radius, and path covering in trees, Cong. Numer. 156 (2002) 155-169.

[53] E. Delavi˜na,R. Pepper, B. Waller, Independence, radius, and hamiltonian paths, MATCH Commun. Math. Comput. Chem. 58(2) (2007) 481-510. [54] E. Delavi˜na,R. Pepper, B. Waller, Lower bounds for the domination number, Discuss. Math. Graph Theory 30(3) (2010) 475-487.

[55] A. A. Dobrynin, A. A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994) 1082-1086.

[56] T. Doˇsli´c,M. Saheli, D. Vukiˇcevi´c,Eccentric connectivity index: extremal graphs and values, Iran. J. Math. Chem. 1 (2) (2010) 45-56.

[57] J.K. Doyle, J.E. Graver, Mean distance in a graph, Discrete Math. bf 17 (1977) 147–154.

[58] Z. Du, B. Zhou, Degree distance of unicyclic graphs, Filomat 24(4) (2010) 95-120.

[59] H. Dureja, S. Gupta, A. K. Madan, Predicting anti-HIV-1 activity of 6-arylbenzonitriles: Computational approach using superaugmented eccentric connectivity topochemical indices, J. Mol. Graph. Model. 26 (2008) 1020-1029.

[60] H. Dureja, A. K. Madan, Topochemical models for the prediction of permeability through blood-brain barrier, Int. J. Pharm. 323 (2006) 27-33.

[61] R. D. Dutton, S. R. Medidi, R.C. Brigham, Changing and unchanging of the radius of a graph, Linear Algebra Appl. 217 (1995) 67–82.

[62] Y. Egawa, K. Inoue, Radius of (2k − 1)-connected graphs, Ars Combin. 51 (1999) 89-95.

[63] R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czech. Math. J. 26 (1976) 283–296.

[64] R.C. Entringer, I. Gutman, A.A. Dobrynin, Wiener index of Trees: Theory and Applica- tions. Acta Appl. Math. 66 (2001) 211-249.

[65] R.C. Entringer, D.J. Kleitman, L.A. Sz´ekely, A note on spanning trees with minimum average distance, Bull. Inst. Combin. Appl. 17 (1996) 71–78.

[66] P. Erd¨os,J. Pach, J. Spencer, On the mean distance between points of a graph, Congr. Numer. 64 (1988) 121-124.

[67] P. Erdös,J. Pach, R. Pollack, Z. Tuza, Radius, diameter and minimum degree, J. Combin. Theory B 47 (1989) 73-79. [68] P. Erdös,A. Rényi, On a problem in the theory of graphs (in Hungarian), Publ. Math. Inst. Acad. Sci. 7 (1962).

[69] P. Erdös,A. Rényi, V.T. Sós,On a problem of graph theory, Studia Sci. Math. Hungar. 1 (1966) 215-235.

[70] P. Erd¨os, M. Saks, V.T. S´os,Maximum induced trees in graphs, J. Combin. Theory B 41(1) (1986) 61-79.

[71] S. Fajtlowicz, A characterization of radius critical graphs, J. Graph Theory 12(4) (1988) 529-533.

[72] S. Fajtlowicz, W. A. Waller, On two conjectures of GRAFFITI I, Congr. Numer. 55 (1986) 51-56.

[73] S. Fajtlowicz, W. A. Waller, On two conjectures of GRAFFITI II, Congr. Numer. 60 (1987) 187-197.

[74] G. Fan, On diameter 2-critical graphs, Discrete Math. 67(3) (1987) 235-240.

[75] O. Favaron, M. Kouider, M. Mah´eo,Edge-vulnerability and mean distance, Networks 19 (1989) 493-504.

[76] L. Feng, W. Liu, The Maximal Gutman index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 66 (2011) 669–708.

[77] Z. F¨uredi,The maximum number of edges in a minimal graph of diameter 2, J. Graph Theory 16 (1992) 81-98.

[78] F. Gliviak, On certain classes of graphs of diameter two without superﬂuous edges, Acta Fac. Rerum Natur. Univ. Comenian. Math. 21 (1968) 39-48.

[79] F. Gliviak, On radially critical graphs, In: Recent Advances in Graph Theory. Proc. Sym- pos. Prague, Academia Praha, Prague (1975), 207-221.

[80] F. Gliviak, Vertex-critical graphs of given diameter, Acta Math. Acad. Sci. Hungar. 27 (1976) 255-262.

[81] F. Gliviak, M. Knor, D. Solt´es,Twoˇ radially maximal graphs with special centers, Math. Solvaca 45(3) (1995) 227-233. [82] F. Gliviak, M. Knor, D. Solt´es,Onˇ radially maximal graphs, Autralas. J. Combin. 9 (1994) 275-284.

[83] D. Goldsmith, B. Manvel, V. Farber, A lower bound for the order of a graph in terms of the diameter and minimum degree, J. Combin. Inform. System Sciences 6 (1981) 315-319.

[84] M. Grover, B. Singh, M. Bakshi, S. Singh, Quantitative structure-property relationships in pharmaceutical research - Part 1, Pharm. Sci. Technol. Today 3 (1) (2000) 28-35.

[85] S Gupta, M. Singh, A. K. Madan, Eccentric distance sum: A novel graph invariant for predicting biological and physical properties, J. Math. Anal. Appl. 275 (2002) 386-401.

[86] S. Gupta, M. Singh, A. K. Madan, Application of Graph Theory: Relationship of Eccentric Connectivity Index and Wieners Index with Anti-inﬂammatory Activity, J. Math. Anal. Appl. 266 (2002) 259–268.

[87] I. Gutman, Selected properties of the Schultz Molecular Topological Index, J. Chem. Inf. Comput. Sci. 34 (1994) 1087–1089.

[88] J. Harant, An upper bound for the radius of a 3-connected planar graph with bounded faces, Contemporary Methods in Graph Theory, 353-358, Bibliographisches Inst., Mannheim 1990.

[89] J. Harant, An upper bound for the radius of a 3-connected graph, Discrete Math. 122(1-3) (1993) 335-341.

[90] J. Harant, H. Walther, On the radius of graphs, J. Combin. Theo. B 30(1) (1981) 113-117.

[91] F. Harary, R. Z. Norman, The dissimilarity characteristic of Husimi trees, Ann. Math. 58 (1953) 134-141.

[92] F. Harary, C. Thomassen, Anticritical graphs, Math. Proc. Cambridge Philos. Soc. 79 (1976) 11-18.

[93] A. Haviar, P. Hrnˇciar,G. Monoszova, Eccentric sequences and cycles in graphs, Acta Univ. M. Belii Ser. Math. no. 11 (2004) 7-25.

[94] A. Haviar, P. Hrnˇciar,G. Monoszov´a,Minimal eccentric sequences with least eccentricity three, Acta Univ. Mathaei Belii Nat. Sci. Ser. Math. No. 5 (1997) 27-50. [95] M.A. Henning, S. Mukwembi, Domination, radius and minimum degree, Discrete Appl. Math. 157(13) (2009) 2964-2968.

[96] S. Hossein-Zadeh, A. Hamzeh, A.R. Ashraﬁ, Extremal properties of Zagreb coindices and degree distance of graphs, Miskolc Math. Notes 11(2) (2010) 129-137.

[97] Y. Hou, A. Chang, The unicyclic graphs with maximum degree distance, J. Math. Study 39 (2006) 18-24.

[98] P. Hrnˇciar,G. Monoszov´a,Minimal eccentric sequences with two values, Acta Univ. M. Belii Ser. Math. No. 12 (2005) 43-65.

[99] H. Hua, K. Xu, S. Wen, A short and uniﬁed proof of Yu et al.’s two results on the eccentric distance sum, J. Math. Anal. Appl. 382 (2011) 364–366.

[100] M. Knor, P. Potocnik, R. Skrekovski, Relationship between edge-wiener and Gutman index of a graph, IMFM 49 (2011) (pre-print).

[101] T. Iida, Order and radius of 3-connected graphs, AKCE Int. J. Graphs Combin. 4(1) (2007) 21-49.

[102] T. Iida, K. Kobayashi, Order and radius of (2k − 1)-connected graphs, SUT J. Math. 42(2) (2006) 335-356.

[103] A. Ili´c,Eccentric connectivity index, (pre-print).

[104] A. Ili´c,D. Stevanovi´c,L. Feng, G. Yu, P. Dankelmann, Degree distance of unicyclic and bicyclic graphs, Discrete Appl. Math. 159 (2011) 779-788.

[105] A. Ili´c,G. Yu, L. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011) 590–600.

[106] K. Inoue, Radius of 3-connected graphs, SUT J. Math. 32(1) (1996) 83-90.

[107] D. S. Johnson, J.K. Lenstra, A.H.G. Rinnooy-Kan, The complexity of the network design problem, Networks 8 (1978) 279-285.

[108] C. Jordan, Sur les assemblages des lignes, J. Reine Angew. Math. 70 (1869) 185-190.

[109] S.-J. Kim, D. G. West, Triangle-free planar graphs with minimum degree at least 3 have radius at least 3, Discuss. Math. Graph Theory 28(3) (2008) 563-566. [110] V. Klee, H. Quaife, Minimum graphs of speciﬁed diameter, connectivity and valence I, Math. Oper. Research 1 (1976) 28-31.

[111] V. Klee, H. Quaife, Classiﬁcation and enumeration of minimum (d, 1, 3)-graphs and (d, 2, 3)-graphs, J. Combin. Theory B 23 (1977) 83-93.

[112] D.J. Klein, Z. Mihalić,D. Plavˇsić,N. Trinajstić,Molecular topological index, a relation with the Wiener index, J. Chem. Inf. Comput. Sci. 32 (1992) 304-305.

[113] M. Kouider, P. Winkler, Mean distance and minimum degree, J. Graph Theory 25 (1997) 95-99.

[114] V. Krishnamoorthy, R. Nandakumar, A class of counterexamples to a conjecture on diameter-critical graphs, Combinatorics and Graph Theory (Calcutta, 1980), pp. 297-300, Lecture Notes in Math., 885, Springer, Berlin-New York, 1981.

[115] R. Laskar, D. Shier, On powers and centers of chordal graphs, Discrete Appl. Math. 6 (1983) 139-147.

[116] V. Lather, A. K. Madan, Models for the prediction of adenosine receptors binding activity of 4-amino(1,2,4)triazolo(4,3-a) quinoxalines, J. Mol. Struct. (THEOCHEM) 678 (2004) 1-9.

[117] V. Lather, A. K. Madan, Topological models for the prediction of anti-HIV activity of dihydro (alkylthio) (naphthylmethyl) oxopyrimidines, Bioorg. Med. Chem. 13 (2005) 1599- 1604.

[118] V. Lather, A. K. Madan, Topological models for the prediction of HIV-protease inhibitory activity of tetrahydropyrimidin-2-ones, J. Mol. Graph. Model. 23 (2005) 339-345.

[119] L. Lesniak-Foster, Eccentric sequences in graphs, Period. Math. Hungar. 6(4) (1975) 287-293.

[120] X. Li, I. Gutman (Eds.), Mathematical aspects of Randic-type molecular structure descriptors, Mathematical Chemistry Monographs, vol 1, University of Kragujevac, Kragujevac (2006).

[121] L. Lovász, Combinatorial Problems and Exercises. AkadémiaiKiadó,Budapest 1979. [122] M. Miller, J. Siráˇn,Mooreˇ graphs and beyond: A survey of the degree-diameter problem, Electronic J. Combin. Dynamic Survey D14 (2005).

[123] M. Miller, G. Pineda-Villavicencio, Complete catalogue of graphs of maximum degree 3 and defect at most 4, Discrete Appl Math. 157(13) (2009) 2983-2996.

[124] M. J. Morgan, S. Mukwembi, H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math. 311 (2011) 1229-1234.

[125] M. J. Morgan, S. Mukwembi, H. C. Swart, A Lower Bound on the Eccentric Connectivity Index of a Graph, Discrete Appl. Math. (2011) doi:10.1016/j.dam.2011.09.010.

[126] M. J. Morgan, S. Mukwembi, H. C. Swart, Extremal Regular Graphs for the Eccentric Connectivity Index, (submitted).

[127] J. W. Moon, On the diameter of a graph, Mich. Math. J. 12 (1965) 349-351.

[128] D. Mubayi, D.B. West, On the number of vertices with speciﬁed eccentricity, Graphs Combin. 16 (2000) 441-452.

[129] S. Mukwembi, On the Gutman Index, (submitted).

[130] S. Mukwembi, The radius of a triangle-free graph with prescribed edge-connectivity, Util. Math. 77 (2008) 135-144.

[131] S. Mukwembi, On diameter and inverse degree of a graph, Discrete Math. 310 (2010) 940-946.

[132] B. R. Myers, The Klee and Quaife minimum (d, 1, 3)-graphs revisited, IEEE Trans. Cir- cuits Syst. 27 (1980) 214-220.

[133] B. R. Myers, The minimum order cubic graphs with speciﬁed diameter, IEEE Trans. Circuits Syst. 27 (1980) 698-709.

[134] R. Nandakumar, On some eccentricity properties of graphs, Ph.D. Thesis, Indian Institute of Technology, India 1986.

[135] R. Nandakumar, K.R. Parthasarathy, Eccentricity-preserving spanning trees, J. Math. Phys. Sci. 24(1) (1990) 33-36. [136] Y. Nishanov, On radially critical graphs with the maximum diameter (in Russian), Trudy Samarkand. Gos. Univ. 235 (1972) 138-147.

[137] Y. Nishanov, A lower bound on the number of edges in radially critical graphs (in Russian), Trudy Samarkand. Gos. Univ. 256 (1975) 77-80.

[138] M. Saheli, A. R. Ashraﬁ, The eccentric connectivity index of armchair polyhex nanotubes, Maced. J. Chem. Chem. Eng. 29 (1) (2010) 71-75.

[139] V. Sharma, R. Goswami, A. K. Madan, Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci. 37(2) (1997) 273-282.

[140] O. Ore, Diameters in graphs, J. Combin. Theory 5 (1968) 75-81.

[141] P. A. Ostrand, Graphs with speciﬁed radius and diameter, Discrete Math. 4 (1973) 71-75.

[142] U.N. Peled, R. Petreschi, A. Sterbini, (n, e)-graphs with maximum sub of squares of degrees, J. Graph Theory 31 (1999) 283-295.

[143] J. Plesnik, Critical graphs of given diameter, Acta Fac. Rerum Natur. Univ. Commenian. Math. 30 (1975) 71-93.

[144] J. Plesn´ık,On the sum of all distances in a Graph or digraph, J. Graph Theory 8 (1984) 1-24.

[145] E. Prisner, Eccentricity-approximating trees in chordal graphs, Discrete Math. 220(1-3) (2000) 263-269.

[146] A. Proskurowski, Centers of maximal outerplanar graphs, J. Graph Theory 4 (1980) 75-79.

[147] G. F. Royle, Regular vertex diameter critical graphs, Australas. J. Combin. 26 (2002) 209-217.

[148] T. Segawa, Radius increase caused by edge-deletion, SUT J. Math. 30(2) (1994) 159-162.

[149] R. Skurnick, Eccentricity sequences of trees and their caterpillar cousins, Graph Theory Notes N. Y. 37 (1999) 20-24.

[150] V. P. Soltan, V. D. Chepoi, d-convex sets in triangulated graphs (in Russian), Mathemat- ical modeling of processes in economics, Mat. Issled. 78 (1984) 105-124. [151] L. Solt´es,Transmission in graphs: A bound and vertex removing, Math. Slovaca 41 (1991) 11-16.

[152] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley–VCH, Weinheim, (2000).

[153] A.I. Tomsecu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 156 (2008) 125-130.

[154] I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. 98 (1999) 159-163.

[155] I. Tomescu, Properties of connected graphs having minimum degree distance, Discrete Math. 309 (2009) 2745-2748.

[156] I. Tomescu, Ordering connected graphs having small degree distances, Discrete Appl. Math. 158 (2010) 1714-1717.

[157] O. Vacek, Diameter-invariant graphs, Math. Bohem. 130(4) (2005) 355-370.

[158] O. Vacek, On critical and co-critical radius edge-invariant graphs, Discuss. Math. Graph Theory 28(3) (2008) 393-418.

[159] O. Vacek, The number of edges of radius-invariant graphs, Math. Slovaca 59(2) (2009) 201-220.

[160] V. G. Vizing, The number of edges in a graph of given radius, Soviet Math. Dokl. 8 (1967) 535-536.

[161] H. B. Walikar, F. Buckley, M. K. Itagi, Radius-edge-invariant and diameter-edgeinvariant graphs, Discrete Math. 272(1) (2003) 119-126.

[162] H. B. Walikar, F. Buckley, M. K. Itagi, Radius-essential edges in a graph, J. Combin. Math. Combin. Comput. 53 (2005) 209-220.

[163] H. Wiener, Structural determination of the paraﬃn boiling points, J. Am. Chem. Soc. 69 (1947) 17-20. [164] B.Y. Wu, G. Lancia, V. Bafna, K.M. Chao, R. Ravi, C.Y. Tang, A polynomial time approximation scheme for minimum routing cost spanning trees, SIAM J. Comput. 29 (2000) 761-778.

[165] H.-G. Yeh, G. J. Chang, Centers and medians of distance-hereditary graphs, Discrete Math. 265(1-3) (2003) 297-310.

[166] Y.-N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete Math. 135 (1994) 359–365.

[167] G. Yu, F. Feng, A. Ili´cOn the eccentric distance sum of trees and unicylic graphs, J. Math. Anal. Appl. 375 (1) (2011) 99-107.

[168] J. Zhang, J. Li, On the maximal eccentric distance sums of graphs, ISRN Applied Mathe- matics (2011), doi:10.5402/2011/421456.

[169] B. Zhou, Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem 63 (2010) 181-198.