arXiv:1510.01903v1 [math.CO] 7 Oct 2015 rga fSaga uiia dcto omsin(No:14ZZ016) Commission Education Municipal Shanghai of Program oicesn euneo ongtv integers nonnegative of sequence nonincreasing A Introduction 1 rlrdu,Wee index. Wiener radius, tral hrceie l rpi eresequences. degree graphic all characterized eresqec n uhagahi alda called is graph a such and sequence degree d 1 ∗ ≥ · · · ≥ M Classifications: AMS words: Key upre yNtoa aua cec onaino hn (No.11 China of Foundation Science Natural National by Supported xrmlGahTer o ereSequences Degree for Theory Graph Extremal h hoai ubrec ngvnst ihtesm re un included tree, are same conjectures the some with Moreover, o sets sequences. number given degree the in etc, index, number Harary chromatic the the index, Wiener the ra invareigenvalue, spectral graph Laplacian) of signless values p-Laplacian, minimum) (Laplacian, graphs (or maximum charac connected and the simple study having we all graphs particular, of In consisting sequence. degree set graphic given a in lems hspprsressm eetrslsadpors nteex the on progress and results recent some surveys paper This d n − 1 scalled is rpi eresqec,mjrzto,te,uiylcgah,s graphs, unicyclic tree, majorization, sequence, degree Graphic 0 ogha od hnhi 020 ..China P.R. 200240, Shanghai, road, Dongchuan 800 eateto ahmtc n MOE-LSC and Mathematics of Department graphic hnhiJa ogUniversity Tong Jiao Shanghai 55,0C5 50,0C2 05C35 05C12, 05C07, 05C05, 05C50, mi:[email protected] Email: ioDn Zhang Xiao-Dong fteeeit ipegahhaving graph simple a exists there if Abstract 1 realization π ( = of is h rtDirichlet first the dius, d 0 d , π . r¨sadGla [17] Gallai Erd¨os and . eieteextremal the terize 1 , · · · 726 n Innovation and 271256) cci,graphic icyclic, ihtesame the with utesand subtrees f at uhas such iants d , rmlprob- tremal n − π 1 with ) sisvertex its as d ∗ pec- 0 ≥ Theorem 1.1 [17] A nonincreasing sequence of nonnegative integers π =(d0,d1, ··· , n−1 dn−1) is graphic degree sequence if and only if i=0 di is even and

r n−1 P di ≤ r(r +1)+ min{r +1,di}, i=0 i=r+1 X X for r =0, 1, ··· , n − 2.

Moreover, Senior [41] and Hakimi[24] characterized all graphic degree sequences with at least a realization being connected.

Theorem 1.2 ([41],[24]) A nonincreasing sequence of nonnegative integers π =(d0,d1,

··· ,dn−1) is graphic sequence with at least a realization being connected if and only if n−1 i=0 di is even and

P r n−1 di ≤ r(r +1)+ min{r, di}, i=0 i=r+1 X X n−1 n(n−1) for r =0, 1, ··· , n − 2, i=0 di ≥ 2 and dn−1 ≥ 1.

Let π be graphic degreeP sequence with at least a realization being connected. Denote by G(π) the set of all connected graphs which are realizations of π, i.e.,

G(π)= {G |G is connected and has π as its degree sequence}. (1)

Sierksma and Hoogeveen [42] surveyed seven criteria for a nonnegative integer se- quence being graphic. For graphic sequences, Ferrara [19] surveyed recent research progress and new results on graphic sequences and presented a number of approach- able open problems. Extremal graph theory is an extremal important branch of graph theory. One is interested in relations among the various graph invariants, such as order, size, connectivity, chromatic number, diameter and eigenvalues, and also in the values of these invariants which ensure that the graph has certain properties. Bollob´as published an excellent book [7] and recent survey in [8]. Erd¨os, Jackson and Lehel in [18] proposed the problem on the possible clique number attained by graphs with the same degree sequence, i.e., determine the smallest integer number M so that

each graphic sequence π = (d0, ··· ,dn−1) with d0 ≥ d1 ≥ ··· ≥ dn−1 ≥ 1 and n−1 i=0 di ≥ M has a realization G having given clique number. Li and Yin [32] P 2 surveyed extremal graph theory and degree sequences, in particular for the above problem and its variants. On the other hand, many graphs coming from the real world have power law degree distribution: the number of vertices with degree k is proportional for k−β for some exponent β ≥ 1. Chung and Lu (for example, see [11], [12]) proposed the famous Chung-Lu’s model: random graphs with given expected degree sequences and studied their properties such as diameters, eigenvalues, the average distance, etc. In this survey, we just focus on some extremal properties of graphs with given degree sequences. Graph invariants such as the spectral radius, the Dirichlet eigenvalues, the Wiener index, the Harary index, the number of subtrees, etc., will be considered. Graphic degree sequences such as tree degree sequences, unicyclic degree sequences etc, will be involved. Moreover, the majorization theorem on two different degree sequences are obtained. Therefore, we may propose the following general problem.

Problem 1.3 For a given graphic degree sequence π, let

Gπ = {G | G is connected with π as its degree sequence}.

For some graph invariants, such as spectral radius, the Wiener index, etc, find the

maximum (minimum) value of these graph invariants in Gπ and characterize all ex- tremal graphs which attain these values.

2 Preliminary

In this section, we introduce some notions and properties. For a graph G = (V, E)

with a root v0, denote by dist(v, v0) the distance between vertices v and v0. Moreover,

the distance dist(v, v0) is called the height h(v)= dist(v, v0) of a vertex v.

Definition 2.1 ([4], [61]) Let G = (V, E) be a graph with root v0. A well-ordering ≺ of the vertices is called a breadth-first search ordering (BFS-ordering for short) if the following holds for all vertices u, v ∈ V : (1) u ≺ v implies h(u) ≤ h(v); (2) u ≺ v implies d(u) ≥ d(v); (3) let uv ∈ E(G), xy ∈ E(G), uy∈ / E(G),xv∈ / E(G) with h(u) = h(x) = h(v) − 1= h(y) − 1. If u ≺ x, then v ≺ y. We call a graph that has a BFS-ordering of its vertices a BFS-graph.

3 For example, for a given degree sequence π = (4, 4, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), ∗ Tπ is the tree of order 17 (see Fig.1). There is a vertex v01 in layer 0; four ver-

tices v11, v12, v13, v14 in layer 1; nine vertices v21, v22, ··· , v29 in layer 2; three vertices

v31, v32, v33 in layer 3.

❝v ✏✏ P01P ✏✏ ❅ PP ✏✏ ❅ PP ✏✏ PP ✏✏ ❅ PP ✏✏ ❅ PP ✏✏✏ ❅ PPP ❝v ❝v ❝v ❝ v ❅ 11 ❅ 12 ❅ 13 ❅ 14 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❝ ❝❅ ❝ ❝ ❅ ❝ ❝ ❅ ❝ ❝ ❅ ❝ ✁ ❆v21 v22 v23 v24 v25 v26 v27 v28 v29 ✁ ❆ ❝✁ ❆ ❝ ❝ v31 v32 v33

Figure 1

It is easy to see that every tree has an ordering such that it satisfies the conditions (1) and (3) by using breadth-first search, but not every tree has a BFS-ordering. For example, the following tree T of order 17 does not have a BFS-ordering with degree sequence π = (4, 4, 3, 3, 3, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (see Fig.2).

❝v ✏✏ P01P ✏✏ ❅ PP ✏✏ ❅ PP ✏✏ PP ✏✏ ❅ PP ✏✏ ❅ PP ✏✏✏ ❅ PPP ❝v ❝v ❝v ❝ v ❅ 11 ❅ 12 ❅ 13 ❅ 14 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❝ ❝❅ ❝ ❝ ❅ ❝ ❝ ❅ ❝ ❝ ❅ ❝ v22 v23 v24✁ ❆ v21 v25 v26 v27 v28 v29 ✁ ❆ ❝ ❝✁ ❆ ❝ v33 v31 v32

Figure 2

Moreover, it is easy to see [4] that there may be many BFS graphs for a given graphic degree sequence. Another importation notion is majorization. Let x =

4 (x1, x2, ··· , xp) and y =(y1,y2, ··· ,yp) be two nonnegative integers. We arrange the entries of x and y in nonincreasing order x↓ =(x[1], ··· x[p]) and y↓ =(y[1], ··· ,y[p]). If

k k

x[i] ≥ y[i], for k =1, ··· , p, (2) i=1 i=1 X X x is said to weakly majorize y and denoted x⊲w y or y⊳w x. Further, if y⊳w x and

p p

x[i] = y[i], (3) i=1 i=1 X X x is said to majorize y and denoted by x⊲y or y ⊳ x. For details, the readers are referred to the book of Marshall and Olkin [35]. It is well known that the following result holds (see [17] or [53]).

′ ′ ′ Proposition 2.2 ([17],[53]) Let π = (d0, ··· dn−1) and π = (d0, ··· ,dn−1) be two different nonincreasing graphic degree sequences. If π ⊳ π′, then there exists a se- ′ ries graphic degree sequences π1, ··· , πk such that π ⊳ π1 ⊳ ··· ⊳ πk ⊳ π , and only two components of πi and πi+1 are different from 1.

Let G =(V, E) be a simple graph with vertex set V (G)= {v1, ··· , vn} and edge set E(G). There are many matrices associated with a graph. Let A(G)=(aij) be the

(0, 1) adjacency matrix of G, where aij = 1 if vi and vj are adjacent, and 0 elsewhere. The spectral radius of A(G) is denoted by ρ(G) and called the spectral radius of G. Moreover, let f be the unit eigenvector of A(G) corresponding to ρ(G). By the Perron-Frobenius theorem, f is unique and positive. So f is called Perron vector of

G. In addition, denote by d(vi) the degree of vertex vi and D(G)= diag(d(u),u ∈ V ) the diagonal matrix of vertex degrees of G, thus the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of a graph G. The spectral radius of L(G) is equal to the largest eigenvalue of L(G) and denoted by λ(G). Moreover, λ(G) is called the Laplacian spectral radius of G. L(G)= D−1/2L(G)D−1/2 is call the normal Laplacian matrix of a graph. The spectral radius of L(G) is denoted by µ(G) and called the normal Laplacian spectral radius. In addition, the matrix Q(G) = D(G)+ A(G) is called the signless Laplacian matrix of G. The spectral radius of Q(G) is denoted by q(G) and called the signless Laplacian spectra radius. Let g be the unit eigenvector of Q(G) corresponding to eigenvalue q(G) and be called signless Laplacian Perron vector of G. The adjacency, Laplacian, normal Laplacian and signless Laplacian matrices play a key role in spectral graph theory. The book “ Spectra of graph”

5 [13] by Cvetkovi´c, Doob abd Sachs focuses on the adjacency matrix. The book “ Applications of combinatorial matrix theory to Laplacian matrices of graphs” [37] by Molitierno studies many properties of Laplacian matrices. The book “ Spectral graph theory” [10] by Chung focuses on normal Laplacian matrix. Moreover, there are several surveys on the topic of signless Lapacian matrices (see [14] and the references therein).

3 Eigenvalues with given tree degree sequences

n−1 If a nonnegative sequence π = (d0, ··· ,dn−1) is graphic and i=0 di = 2(n − 1), π T is called tree degree sequence and denoted by π the set of allP trees having degree sequence π.

For a tree degree sequence π = (d0,d1, ··· ,dn−1) with d0 ≥ d1 ≥ ··· ≥ dn−1, ∗ With the aid of breadth-first search method, we can define a special tree Tπ with

degree sequence π as follows. Assume that dm > 1 and dm+1 = ··· = dn−1 = 1 for

0 ≤ m < n − 1. Put s0 = 0. Select a vertex v01 as a root and begin with v01 in layer 0. Put s1 = d0 and select s1 vertices {v11, ··· , v1,s1 } in layer 1 such that they are adjacent to v01. Thus d(v01)= s1 = d0. We continue to construct all other layers by recursion. In general, put st = ds0+s1+···+st−2+1 + ··· + ds0+s1+···+st−2+st−1 − st−1 for t ≥ 2 and assume that all vertices in layer t have been constructed and are denoted by {vt1, ··· , vtst } with d(vt−1,1)= ds0+···+st−2+1, ··· ,d(vt−1,st−1 )= ds0+···+st−1 . Now using the induction hypothesis, we construct all vertices in layer t + 1. Put st+1 = ds0+···+st−1+1+···+ds0+···+st −st. Select st+1 vertices {vt+1,1, ··· vt+1,st+1 } in layer t+1 such that vt+1,i is adjacent to vtr for r = 1and1 ≤ i ≤ ds0+···+st−1+1−1and for2 ≤ r ≤ st and ds0+···+st−1+1 +ds0+···+st−1+2 +···+ds0+···st−1+r−1 −r+2 ≤ i ≤ ds0+···+st−1+1 + ds0+···+st−1+2 + ··· + ds0+···+st−1+r − r. Thus d(vtr) = ds0+···+st−1+r for 1 ≤ r ≤ st.

Assume that m = s0+···+sp−1+q. Put sp+1 = ds0+···+sp−1+1+···+ds0+···+sp−1+q−q and select sp+1 vertices {vp+1,1, ··· , vp+1,sp+1 } in layer p + 1 such that vp+1,i is adjacent to vpr for 1 ≤ r ≤ q and ds0+···+st−1+1+···+ds0+···+sp−1+2+···+ds0+···sp−1+r−1−r+2 ≤ i ≤ ds0+···+sp−1+1 + ···+ ds0+···+sp−1+2 + ···+ ds0+···+sp−1+r − r. Thus d(vp,i)= ds0+···+sp−1+i ∗ ∗ for 1 ≤ i ≤ q. In this way, we obtain a tree Tπ . It is easy to see that Tπ has degree ∗ sequence π. Moreover, this special tree Tπ is called greedy tree with a given tree degree sequence π. In addition, it is easy to show the following assertion holds.

Proposition 3.1 ([4], [60]) Given a tree degree sequence π, there exists a unique

6 ∗ tree Tπ with degree sequence π having a BFS-ordering. In other words, the greedy tree

with π is only tree having a BFS-ordering in the set Tπ. Moreover, any two trees with the same degree sequences and having BFS-ordering are isomorphic.

3.1 The (Laplacian, signless Laplacian) spectral radius

Biyiko˘glu and Leydold [4]characterized all extremal trees having the largest spectral

radius in the set Tπ consisting of all trees with a given degree sequence π. While, Zhang [60] characterized all extremal trees having the largest (signless) Laplacian

spectral radius in the set Tπ consisting of all trees with a given degree sequence π, since the signless Laplacian radius of a tree T is always equal to its the Laplacian spectral radius. It is not a surprise that the two extremal graphs are coincident.

∗ Theorem 3.2 ([4], [60]) For a given tree degree sequence π, the greedy tree Tπ is

the only tree having the largest spectral radius (resp. signless Laplacian radius) in Tπ. Moreover, The BFS-ordering of the greedy tree is consistent with the Perron vector f of G in such a way that f(u) > f(v) (resp. g(u) >g(v)) implies u ≺ v.

Theorem 3.3 ([4], [60]) Let π and τ be two different tree degree sequences with the ∗ ∗ same order. Let Tπ and Tτ have the largest spectral radii (resp. Laplacian, signless ∗ ∗ Laplacian spectral radii) in Tπ and Tτ , respectively. If π ⊳ τ, then ρ(Tπ ) < ρ(Tτ ), ∗ ∗ ∗ ∗ λ(Tπ ) <λ(Tτ ) and q(Tπ ) < q(Tτ ). With the aid of Theorems 3.2 and 3.3, we may deduce extremal graphs with the largest (resp. signless Laplacian) spectral radius in some class of graphs. (1) Corollary 3.4 ([4],[60]) Let Tn,s be the set of all trees of order n with s leaves and let ∗ Tπ be the greedy tree with π =(t, 2, ··· , 2, 1, ··· , 1) (i.e., if n − 1= sq + t, 0 ≤ t

7 (3) Corollary 3.6 ([59],[60]) Let Tn,α be the set of all trees of order n with the inde- ∗ pendence number α and let Tπ be the greedy tree with π = (α, 2, ··· , 2, 1, ··· , 1) the ∗ ∗ numbers n − α − 1 of 2 and α of 1. Then ρ(T ) ≤ ρ(Tπ ) (resp. λ(T ) ≤ λ(Tπ ) , ∗ (3) ∗ q(T ) ≤ q(Tπ )) for any tree T ∈ Tn,α , with equality if and only if T = Tπ .

(4) Corollary 3.7 ([22], [60]) Let Tn,β be the set of all trees of order n with the matching ∗ number β and Tπ be the greedy tree with π =(n−β, 2, ··· , 2, 1, ··· , 1) and the number ∗ ∗ ∗ n − β of 1. Then ρ(T ) ≤ ρ(Tπ ) (resp. λ(T ) ≤ λ(Tπ ) , q(T ) ≤ q(Tπ )) for any tree (4) ∗ T ∈ Tn,β , with equality if and only if T = Tπ .

On the minimum (Laplacian, signless Laplacian) spectral radius of with a given tree degree sequence, it seems to be more difficult. The related results are referred to [6].

3.2 The spectral radius of p-Laplacian of weighted trees

Now we turn to consider p-Laplacian eigenvalues of weighted graphs. Let GW =

(V (G), E(G), W (G)) be a weighted graph with vertex set V (G) = {v0, v1, ··· , vn−1}, edge set E(G) and weight set W (G)= {wk > 0,k =1, 2, ··· , |E(G)|}. If uv ∈ E(G), denote by wG(uv) the weight of an edge uv; if uv∈ / E(G), define wG(uv)=0. The weight of a vertex u, denoted by wG(u), is the sum of weights of all edges incident to u in G.

For p > 1, the discrete p-Laplacian △p(G) of a function f on V (G) is defined to be [p−1] △p(G)f(u)= (f(u) − f(v)) wG(uv), (4) v,uvX∈E(G) [q] q where x = sign(x)|x| . When p =2, △2(G) is the well-known graph Laplacian (see

[5]), i.e., ∆2(G) = L(G) = D(G) − A(G), where A(G)=(wG(vivj))n×n denotes the weighted adjacency matrix of G and D(G) = diag(wG(v0), wG(v1), ··· , wG(vn−1)) denotes the weighted diagonal matrix of G.

A real number λ is called an eigenvalue of △p(G) if there exists a function f =06 on V (G) such that for u ∈ V (G),

[p−1] ∆p(G)f(u)= λf(u) . (5)

The function f is called the eigenfunction corresponding to λ. The largest eigenvalue p of ∆p(G), denoted by λ (G), is called the p-Laplacian spectral radius.

8 Given a tree degree sequence π and a positive set W , denote by Tπ,W the set of trees with a given tree degree sequence π and a positive weight set W . Thus the ∗ un-weighted greedy tree Tπ has a BFS-ordering v0 ≺ v2 ···≺ vn−1. Hence the unique ∗ ∗ weighted greedy tree Tπ,W is obtained from the un-weighted greedy tree Tπ whose edge

weighted has the following property“ for any two edges vivj and vkvl with i

weight w(vivj) ≥ w(vkvl) if i

Theorem 3.8 [55] For a given tree degree sequence π and a positive weight set W , ∗ the weighted greedy tree Tπ,W is the unique weighted tree with the largest p-Laplacian

spectral radius in Tπ,W , which is independent of p.

∗ Theorem 3.9 [55] Let π and τ be two different tree degree sequences. Let Tπ,W ∗ and Tτ,W be two weighted greedy trees in Tπ,W and Tτ,W respectively. If π ⊳ τ, then p ∗ p ∗ λ (Tπ,W ) <λ (Tτ,W ).

Remark If the positive set W consisting of all 1, Theorems 3.8 and 3.9 become Biyiko˘glu and Leydold’s result [5]. If p = 2, Theorems 3.8 and 3.9 become Tan’s result [48]. If the positive set W consisting of all 1 and p = 2, Theorems 3.8 and 3.9 become Zhang’ result[60]. In addition, Corollaries 3.4, 3.5, 3.6 and 3.7 can be easily generalized to the weighted trees, respectively. Moreover, the p-adjacency eigenvalue may be similarly defined and Theorems 3.8 and 3.9 can be translated to the p-adjacency eigenvalue.

3.3 The Dirichlet eigenvalue of trees with boundary

In this subsection, we consider the Dirichlet eigenvalues of trees with boundary. A graph with boundary G =(V0∪∂V , E0∪∂E) consists of interior vertex set V0, boundary vertex set ∂V , interior edge set E0 that connect interior vertices, and boundary edge set ∂E that join interior vertices with boundary vertices (for example, see [10] or [21]). Throughout this subsection, we always assume that the degree of any boundary vertex is 1 and the degree of any interior vertex is at least 2. A real number λD is called a Dirichlet eigenvalue of G if there exists a function f =6 0 such that they satisfy the Dirichlet eigenvalue problem:

L(G)f(u)= λf(u) u ∈ V0; ( f(u)=0 u ∈ ∂V.

9 The function f is called an eigenfunction corresponding to λD. Recently, Bıyıko˘glu and Leydold [3] proposed the following problem:

Problem 3.10 ([3]) Give a characterization of all graphs in a given class C with the Faber-Krahn property, i.e., characterize those graphs in C which have minimal first Dirichlet eigenvalue for a given “volume”.

In order to study the above problem, Bıyıko˘glu and Leydold [3] extended the con- cept of an SLO*-ordering for describing the trees with the Faber-Krahn property, which was introduced by Pruss (see [38]). The notation of an SLO*-ordering may be extended for any connected graphs.

Definition 3.11 ([3]) Let G =(V0 ∪ ∂V,E0 ∪ ∂E) be a connected graph with root v0. Then a well-ordering ≺ of the vertices is called spiral-like (SLO*-ordering for short) if the following holds for all vertices u,v,x,y ∈ V0 ∪ ∂V :

(1) v ≺ u implies h(v) ≤ h(u), where h(v) denotes the distance between v and v0; (2) Let uv ∈ E(G), xy ∈ E(G),uy∈ / E(G), xv∈ / E(G) with h(u)= h(v) − 1 and h(x)= h(y) − 1. If u ≺ x, then v ≺ y ; (3) If v ≺ u and v ∈ ∂V , then u ∈ ∂V .

(4) If v ≺ u and v,u ∈ V0, then d(v) ≤ d(u).

Clearly, if G is a tree, an SLO*-ordering of G is consistent with the definition of an SLO*-ordering in [3]. Moreover, if there exists a positive integer r such that the number of vertices v with h(v)= i + 1 is not less than the number of vertices v with h(v)= i for i =1, ··· ,r − 1, and h(u) ∈{r, r +1} for any boundary vertex u ∈ ∂V ,

G is called a ball approximation. Given a tree degree sequence π, denote by Tπ,B the set consisting of all trees with boundary and degree sequence π. Bıyıko˘glu and

Leydold [3] proved that Tπ,B contains an SLO*-tree that is uniquely determined up ∗ to isomorphism. This special tree in Tπ,B is denoted by Tπ,B.

Theorem 3.12 [3] For a given tree degree sequence π,

D D ∗ λ (T ) ≥ λ (Tπ,B) for T ∈ Tπ,B

∗ with equality if and only if T = Tπ,B.

Theorem 3.13 [3] Let π and τ be two different tree degree sequences. If π ⊳ τ, then D ∗ D ∗ λ (Tπ,B) >λ (Tτ,B).

10 4 Chemical indices for tree degree sequences

There are many indices which are used to describe molecules and molecular com- pounds in chemical graph theory. One of the most widely known topological de- scriptor is the Wiener index which is named after chemist Wiener [54] who firstly considered it. The Wiener index of a graph G is defined as

W (G)= d(vi, vj). (6) {vi,vXj }⊆V (G) Let T =(V, E) be a rooted tree with root r. For each vertex u, let T (u) be the subtree of the rooted tree T induced by u and all its successors in T . In other words, if u is not the root r of tree T and v is the parent of u, then T (u) is the connected component of T obtained from T by deleting the edge uv such that the component does not contain the root r; if u is the root r, then T (u) is the tree T . Let φT (u)= |T (u)| be

the number of vertices in T (u) and denote φ(T )=(φT (u),u ∈ V (T )). We prove the following results

Theorem 4.1 [64] Let T be a rooted tree in Tπ. Then the following conditions are equivalent: (1) T has a BFS-ordering; ∗ (2) φ(T )↓ = φ(Tπ )↓; ∗ (3) T is isomorphic to Tπ . Further, this result can be used to characterize the trees having the minimum Wiener index among all trees with a given degree sequence. Wang [52] independently proved the same result by a different approach.

∗ Theorem 4.2 ([52],[64]) Given a tree degree sequence π, the greedy tree Tπ is a

unique tree with the minimum Wiener index in Tπ.

∗ Theorem 4.3 [64] Let π and τ be two different tree degree sequences. Let Tπ and ∗ ∗ ∗ Tτ be two the greedy trees in Tπ and Tτ , respectively. If π ⊳ τ, then W (Tπ ) < W (Tτ ).

On the other hand, it is natural to investigate the extremal trees which attain the maximum Wiener index among all trees with given degree sequences. But this prob- lem seems to be difficult, since it is discovered that the extremal trees are not unique and depend on the values of components of degree sequences. However, there are some partial results. A caterpillar is a tree with the property that a path remains if all leaves are deleted and this path is called spine.

11 ♯ Theorem 4.4 [43] If a tree Tπ has the maximum Wiener index among all trees in ♯ Tπ for a given tree degree sequence π, then Tπ is a caterpillar.

Theorem 4.5 [63] Let π = (d0, ··· ,dn−1) with d0 ≥···≥ dk−1 ≥ 2 ≥ dk+1 = ··· = ♯ dn−1 = 1. If a caterpillar Tπ has the maximum Wiener index among all trees in Tπ and the spine is v0v1 ··· vk−1 with d(v0) ≥ d(vk−1), then there exists a 1 ≤ t ≤ k − 2 such that d(v0) ≥ d(v1) ≥···≥ d(vt) and d(vt) ≤ d(vt+1) ≤···≤ d(vk−1).

Further, Schmuck, Wagner and Wang [40] proposed and studied a new graph invariant which are based on distance.

Theorem 4.6 [40] Let π = (d0,d1, ··· ,dn−1) be a tree degree sequence let r be an arbitrary positive integer. If pr(T ) is the number of pairs (u, v) of vertices such that ∗ d(u, v) ≤ r for a tree T ∈ Tπ, then pr(T ) ≤ pr(Tπ ) with equality for all r =1, ··· , n−1 ∗ if and only if T is the greedy tree Tπ .

Recently, Schmuck, Wagner and Wang [40] proposed a general graph invariant

Wψ(T )= ψ(d(u, v)) (7) {u,vX}⊆V (G) for any nonnegative function ψ. Further, they proved the following result.

Theorem 4.7 [40][51] Let π be a tree degree sequence and ψ(x) be any nonnegative and nondecreasing (resp. nonincreasing) function on x. Then for any tree T ∈ Tπ,

∗ Wψ(T ) ≥ (resp. ≤) Wψ(Tπ ). (8)

Further, if ψ(x) is strictly increasing function, then equality holds if and only if T is ∗ the greedy tree Tπ .

x(x+1) Remark. If ψ(x)= x, then Wψ(T ) is the classical Wiener index. If ψ(x)= 2 , 1 then Wψ(T ) is the hyper-Wiener index [29]. If ψ(x) = x , then Wψ(T ) is the Harary index. Moreover, it is easy to deduce the following result from [40] and [51].

Theorem 4.8 Let π and τ be two different tree degree sequences with π ⊳ τ. If ψ(x) is any nonnegative and strictly increasing (resp. decreasing) function on x, then

∗ ∗ Wψ(Tπ ) > (resp. <) Wψ(Tτ ). (9)

12 Proof. Let pr(T ) be the number of pairs {u, v} of vertices with d(u, v) ≤ r. By ∗ ∗ Theorem in [51], pr(Tπ ) ≤ pr(Tτ ) for r = 1, ··· , n with strict inequality for at least one r. Observe that n(n − 1) W (T ∗)= (ψ(k + 1) − ψ(k))( − p (T ∗)) ψ π 2 k π Xk≥0 and n(n − 1) W (T ∗)= (ψ(k + 1) − ψ(k))( − p (T ∗)). ψ τ 2 k τ Xk≥0 ∗ ∗ If ψ(x) is nonnegative and strictly increasing, then Wψ(Tπ ) > Wψ(Tτ ).

Theorem 4.9 [40] Let π be a tree degree sequence and ψ(x) be strictly increasing ♯ and convex function on x. If Tπ is a tree that maximizes Wψ(T ) among all trees in ♯ the set Tπ, then Tπ is a caterpillar.

In order to study total π-electron energy on molecular structure, Gutman and Trinajsti´c[23] proposed the second Zagreb index, which is defined to be

ZA(T )= d(u)d(v), (10) uvX∈E(T ) where d(u) and d(v) are the degrees of vertices u and v, respectively.

∗ Theorem 4.10 [40] Let π be a tree degree sequence. Then the greedy tree Tπ is the only tree having the maximum second zagreb index among all trees in T ∈ Tπ, i.e., for any tree T ∈ Tπ, ∗ ZA(T ) ≤ ZA(Tπ ) (11)

∗ with equality if and only if T is the greedy tree Tπ .

Let G be a simple graph. Denoted by m(G,k) the number of k−matchings in G and m(G, 0) = 1. The Hosoya index, named after Haruo Hosoya, is defined to be

Z(G)= m(G,k) (12) Xk≥0 Denote by i(G,k) the number of k−independent sets in G and i(G, 0) = 1. The Merrifield-Simmons index is defined to be

MS(G)= i(G,k) (13) Xk≥0 13 The energy of graph is defined by the sum of the absolute values of all the eigenvalues of A(G), and denoted by En(G). Andriantiana [1] proved that the greedy tree has minimum energy, Hosoya index and maximum Merrifield-Simmons index.

∗ Theorem 4.11 [1] Let π be a tree degree sequence. Then the greedy tree Tπ is the only tree having minimum energy, minimum Hosoya index and maximum Merrifield-

Simmons indices, respectively among all trees in T ∈ Tπ, i.e., for any tree T ∈ Tπ,

∗ ∗ ∗ Z(T ) ≥ Z(Tπ ), En(T ) ≥ En(Tπ ), MS(T ) ≤ MS(Tπ ), (14)

∗ with any one equality if and only if T is the greedy tree Tπ .

Theorem 4.12 [1] Let π and τ be two different tree degree sequences. Then the ∗ ∗ greedy trees Tπ and Tτ are the minimum energy, minimum Hosoya index and maxi-

mum Merrifield-Simmons indices in Tπ and Tτ respectively. If π ⊳ τ, then

∗ ∗ ∗ ∗ ∗ ∗ En(Tπ ) > En(Tτ ), Z(Tπ ) >Z(Tτ ), MS(Tπ ) < MS(Tτ ). (15)

In addition, there are the following results on the number of subtrees of a tree.

∗ Theorem 4.13 [65] With a given degree sequence π, Tπ is the unique tree with the

maximum number of subtrees in Tπ.

Theorem 4.14 [65] Given two different degree sequences π and τ. If π ⊳ τ, then the ∗ ∗ number of subtrees of Tπ is less than the number of subtrees of Tτ .

5 Unicyclic degree sequences

For a given nonincreasing degree sequence π =(d0,d1, ··· ,dn−1) of a unicyclic graph ∗ with n ≥ 3, we construct a special unicyclic graph Uπ as follows: If d0 = 2, denote ∗ ∗ by Uπ the cycle of order n. If d0 ≥ 3, denote by Uπ is the graph obtained from a ∗ cycle C3 of order 3 and by breadth-first search ordering, i.e., Uπ is BFS graph with the first three vertices, each pair of which is adjacent. Clearly, there exists only one ∗ such unicyclic graph Uπ for a given unicyclic degree sequence π, which is called the greedy unicyclic graph with a given unicyclic degree sequence π.

Proposition 5.1 ([24], [41]) Let π = (d0, ··· ,dn−1) be a positive nonincreasing in- n−1 teger sequence with n ≥ 3. Then π is unicyclic graphic if and only if i=0 di = 2n d ≥ and 2 2. P 14 Theorem 5.2 ([33], [61]) Let π = (d0,d1, ··· ,dn−1) be a unicyclic degree sequence. ∗ Then the greedy unicyclic graph Uπ is the only unicyclic graph having the largest

(signless Laplacian) spectral radius in Uπ, i.e.,

∗ ∗ ρ(G) ≤ ρ(Uπ ), q(G) ≤ q(Uπ ) (16)

∗ with one equality if and only if G is the greedy unicyclic graph Uπ

Theorem 5.3 ([33], [61]) Let π and τ be two different degree sequences of unicyclic ∗ ∗ ∗ ∗ graphs with the same order. If π ⊳ τ then ρ(Uπ ) < ρ(Uτ ) and q(Uπ ) < q(Uτ )

In order to present the results on the Dirichlet eigenvlaues, we need the following

notion. Let π =(d0, ··· ,dk−1, 1, ··· , 1) be a unicyclic degree sequence with d0 ≥ d1 ≥ ∗ ... ≥ dk−1. If d2 ≥ 3 , then let Uπ,B have SLO*-ordering with each pair of the first

three vertices being adjacent; if d0 = ... = dm−1 = 2 and dm =3 for 3 ≤ m ≤ k − 1, ∗ then let Uπ,B be obtained from by identifying one end vertex of a path Pm with one ∗ vertex of triangle and the other end vertex of Pm with the root of Tτ,B having SLO*- ordering with τ =(dm+1 − 1,dm+2, ··· ,dn−1); if d0 = ... = dm−1 = 2 and dm ≥ 4 for ∗ 3 ≤ m ≤ k − 1, then let Uπ,B be obtained from by identifying one vertex of a cycle ∗ Cm and the root of Tτ,B having SLO*-ordering with τ = (dm − 2,dm+2, ··· ,dn−1). ∗ Clearly, Uπ,B is uniquely determined by π. Then we have the following results

Theorem 5.4 [56] For a given graphic unicyclic degree sequence π =(d0,d1,...,dn−1), with 3 ≤ d0 ≤ ... ≤ dk and dk+1 = ··· = dn−1 = 1, let G = (V0 ∪ ∂V,E0 ∪ ∂E) be a

graph with the Faber-Krahn property in Uπ. Then G has an SLO*-ordering consistent with the first eigenfunction f of G in such a way that v ≺ u implies f(v) ≥ f(u).

Theorem 5.5 [56] For a given graphic unicyclic degree sequence π =(d0,d1,...,dn−1) D D ∗ with 3 ≤ d0 ≤ ... ≤ dk and dk+1 = ... = dn−1 = 1, then λ (G) ≥ λ (Uπ,B) with ∗ ∗ equality if and only if G = Uπ,B. In other words, Uπ,B is the unique uncyclic graph with the Faber-Krahn property in Uπ, which can be regarded as ball approximation.

For d0 = 2, we proposed the following conjecture.

Conjecture 5.6 [56] Let π =(d0,d1,...,dk−1, 1,..., 1) be a graphic unicyclic degree D sequence with 2 ≤ d0 ≤ d1 ≤ ... ≤ dk−1 and dk = ... = dn−1 = 1. Then λ (G) ≥ D ∗ ∗ ∗ λ (Uπ,B) with equality if and only if G = Uπ,B. In other words, Uπ,B is the unique

graph with the Faber-Krahn property in Uπ.

15 ∗ Theorem 5.7 [58] For a given positive integer k ≤ n − 3, let Un,k be unicyclic graph obtained from by identifying one end vertex of a path Pn−k−1 and one vertex of a triangle and the other end vertex of Pn−k−1 and the center of star K1,k. Then for any D D ∗ unicyclic graph G of order n with k pendant vertices, λ (G) ≥ λ (Un,k) with equality ∗ ∗ if and only if G = Un,k. In other words, U is the only unicyclic graph with the

Faber-Krahn property in the set Uk of all unicyclic graphs of order n with k pendant vertices, which can be regarded a ball.

6 Graphic degree sequences

In this section, we discuss some extremal properties of Gπ.

Theorem 6.1 ([4],[61])For a given graphic degree sequence π, if G is a simple con-

nected graph that has the largest (resp. signless Laplacian) spectral radius in Gπ, then G has a BFS-ordering, i.e., G is a BFS graph.

Moreover, all extremal graphs having the (signless Laplacian) spectral radius for bicyclic (tricyclic) degree sequence have been characterized (see [26],[28], [33]). On the Dirichlet eigenvalues for bicyclic degree sequences, we have the following

results. For given a graphic bicyclic degree sequence π =(d0, ··· ,dn−1) with 3 ≤ d0 ≤ ∗ ··· dk−1 and dk = ··· = dn−1 = 1, let Gπ,B be the unique graph having SLO* ordering

v0v1 ··· vn−1 with the induced subgraph by {v0, v1, v2, v3} being K4 − v2v3. Then

Theorem 6.2 [57] Let π = (d0,d1, ··· ,dk−1, 1, 1, ··· , 1) be a graphic bicyclic degree

sequence with 3 ≤ d0 ≤ d1 ···≤ dk−1 for k ≥ 4. Then for any bicyclic graph G with D D ∗ degree sequence π, λ (G) ≥ λ (Gπ,B) with equality if and only if G = Gπ,B. In other ∗ words, Gπ,B is the only extremal graph with the smallest first Dirichlet eigenvalue in

Gπ.

′ ′ ′ Theorem 6.3 [57] Let π = (d0,d1, ··· ,dk−1, 1, 1, ··· , 1) and τ = (d0,d1, ··· ,dk−1,

1, 1, ··· , 1) be two different graphic bicyclic degree sequences. If π ⊳ τ with d0 ≥ 3, ′ D D d0 ≥ 3, then λ (Gπ,B) >λ (Gτ,B).

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. An H minor is a minor isomorphic to H. If H is a complete graph, we say that G contains a clique minor of size |H|. For a graph G, the Hadwiger

16 number h(G) of G is the maximum integer k such that G contains a clique minor of size k. Moreover, denote by χ(G) the chromatic number of G. Let

h(π) = max{h(G): G ∈ Gπ}, χ(π) = max{χ(G): G ∈ Gπ}. (17)

Conjecture 6.4 (Hadwigers Conjecture for Degree Sequences, [39]) For every graphic degree sequence π, χ(π) ≤ h(π).

Recently, Dvo˘r´ak and Mohar proved a stronger result than the above conjecture.

Theorem 6.5 [16] For every graphic degree sequence π,

χ(π) ≤ h′(π) ≤ h(π), (18)

′ where h (G) is the maximum k such that G has a Kk- topological minor and χ′(π)= ′ max{h (G): G ∈ Gπ}.

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