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Dankelmann-The Distance Concept and Distances in Graphs

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Dankelmann-The Distance Concept and Distances in Graphs 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 in- vestigations 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 struc- tures [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) = maxu2V (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 Erd¨osand R´enyi [68], Erd¨os, R´enyi and S´os[69], and by Bollob´as[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 Bollob´as and Harary [24], and solved by Bollob´as[17]. 2.2 Diameter and degrees Since the graph that maximises the diameter among all connected graphs, the path, has mini- mum 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], Erd¨os,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 Erd¨oset 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 d^is the harmonic mean of the v2V (G) deg(v) d^ vertex degrees of G.
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