Vertex Degrees and Doubly Stochastic Graph Matrices

Transcription

1 Vertex Degrees and Doubly Stochastic Graph Matrices Xiao-Dong Zhang DEPARTMENT OF MATHEMATICS, SHANGHAI JIAO TONG UNIVERSITY 800 DONGCHUAN ROAD, SHANGHAI, , P. R. CHINA Received May 2, 2006; Revised December 2, 2009 Published online 2 November 200 in Wiley Online Library (wileyonlinelibrary.com). DOI 0.002/jgt Abstract: In this article, the relationship between vertex degrees and entries of the doubly stochastic graph matrix has been investigated. In particular, we present an upper bound for the main diagonal entries of a doubly stochastic graph matrix and investigate the relations between a kind of distance for graph vertices and the vertex degrees. These results are used to answer in negative Merris question on doubly stochastic graph matrices. These results may also be used to establish relations between graph structure and entries of doubly stochastic graph matrices. 200 Wiley Periodicals, Inc. J Graph Theory 66: 04 4, 20 Contract grant sponsor: National Natural Science Foundation of China; Contract grant numbers: 09737; ; Contract grant sponsor: National Basic Research Program (973) of China; Contract grant number: 2006CB805900; Contract grant sponsor: National High Technology Research and Development Program (863) of China; Contract grant number: 2006AAZ209; Contract grant sponsor: Science and Technology Commission of Shanghai Municipality; Contract grant number: 09XD Journal of Graph Theory 200 Wiley Periodicals, Inc. 04

2 VERTEX DEGREES 05 MSC 2000: 05 C 50; 05 C 05; 05 C 2 Keywords: tree; doubly stochastic matrix; distance. INTRODUCTION Let G=(V,E) be a simple graph with vertex set V(G)={v,...,v n } and edge set E(G). Denote by d(v i )ord i the degree of vertex v i. Let D(G)=diag(d,...,d n )bethedegree diagonal matrix and A(G) bethen n adjacency matrix whose (i, j)-entry is if (v i,v j ) E and 0 otherwise. The matrix L(G)=D(G) A(G) is called the Laplacian matrix of G, which may be dated back to Kirchhoff s Matrix-Tree Theorem and has been extensively studied for the past fifty years (e.g. see [, 4] and the references therein). Let I n be the n n identity matrix. It follows from [7] (also see [2]) that Ω(G)=(I n +L(G)) =(ω ij ) is a doubly stochastic matrix. So Ω(G) iscalledthedoubly stochastic graph matrix, which was introduced by Golender et al. [7] (see also [4]) in their study of chemical information processing. Chebotarev and Shamis in [4] pointed out that the doubly stochastic graph matrix may be regarded as the matrix of relative forest accessibility of the vertices of G by the matrix-forest theorem. Therefore these values can be used to measure the proximity among vertices and may be used to evaluate the group cohesion in the construction of sociometric indices. Chebotarev and Shamis [3] and Merris [2] independently proved the important result on the relationship between the entry of the doubly stochastic matrix of a graph and its number of spanning forests. Theorem. (Chebotarev and Shamis [3], Merris [2]). Let G be a simple graph with vertex set V(G)={v,...,v n }.LetF be the set of all spanning forests of G and F(i,j) be the set of all spanning forests of G with both vertices v i and v j belonging to the same component. For F F, define γ(f) to be the product of the numbers of vertices in the connected components of F and γ i (F) to be the product of the numbers of vertices in the connected components of F that do not contain vertex v i.then F F(i,j) ω ij = γ i(f) F F γ(f). It is noted that the formulation in Theorem. had been mentioned in [2] and had been also proved in [5]. Moreover, Merris in [3] established the relationship between the entry ω ij of Ω(G) and structure of G. Theorem.2 (Merris [3]). Let G be a simple graph on n vertices. If ω ij <4/ (n 2 +4n), then v i is not adjacent to v j. Moreover, ω ij is also relative to the weights of routes of various lengths between v i and v j (see [4], Proposition 0). Recently, Zhang in [8] presented some relations between the diameter of a tree and the smallest entry ω(g)=min{ω ij, i,j n}. Roughly speaking, the smaller the ω(g) is, the larger the diameter. Since the algebraic connectivity α(g) of G, i.e. the second smallest eigenvalue of L(G), is related to many graph invariants (see [, 4]), the entries of Ω(G) and related parameters may be further

3 06 JOURNAL OF GRAPH THEORY considered with the aid of the second largest eigenvalue of Ω(G) being /(+α(g)). For more related results, the reader may be referred to [, 5, 6, 0, 6, 7]. Chebotarev et al. in [4] introduced the following notation. Definition.3. Let G=(V,E) be a simple graph on vertex set V ={v,...,v n } with doubly stochastic graph matrix Ω(G)=(ω ij ). Define ρ(v i,v j )=ω ii +ω jj 2ω ij. Chebotarev et al. showed that ρ is a distance and adding an edge to G cannot increase the distance between any pair of its vertices. By Theorem., ρ(v i,v j ) takes into account all connecting paths between vertices v i and v j. Merris in [3] introduced the following notation. Definition.4. Let G be a simple graph on vertex set V ={v,...,v n } with doubly stochastic graph matrix Ω(G)=(ω ij ). Let r(i)= n ρ(v i,v j ) and define j= r(g)=min{r(i), i n}, r(g)=max{r(i), i n}. If r(k)=r(g), then v k is called a most remote vertex; andifr(k)=r(g), then v k is called a least remote vertex. Merris in [3] proved that v k is a most remote vertex if and only if ω kk is a main diagonal maximum entry, and v i is a least remote vertex if and only if ω ii is a main diagonal minimum entry. The usual distance σ(v i,v j ) is the number of edges in the shortest path from v i to v j.themean distance (see [9]) in a connected graph G on n vertices is given by μ(g)=(/n(n )) v i,v j V σ(v i,v j ), which is related to the Wiener index (see []). From [0], it is easy to see that if T is a tree, then μ(t)= 2 n n i=2 λ i λ i, where =λ λ 2 λ n are all eigenvalues of Ω(G). On the other hand, the quantity W ρ (G)= v n(n ) i,v j ρ(v i,v j )= ni= r(i) n(n ) satisfies W ρ (G)= n λ i. n Furthermore, Merris [3] proposed the following question. Question.5. Let G be a simple graph on vertex set V ={v,...,v n } with doubly stochastic graph matrix Ω(G)=(ω ij ).Doesd k >d i for all i k imply r(k)=r(g)? i=2

4 VERTEX DEGREES 07 The following main result of this article consists of two parts. We first answer Question.5 in negative and then prove a sufficient condition under which Question.5 is indeed true. Theorem.6. (i) There exists a family of graphs with d k >d i for all i k butr(k)>r(g). (ii) Let G be a simple graph. If d k 2d i for all i k, then r(k)=r(g). The rest of this article is organized as follows: In Section 2, we present an upper bound for the main diagonal entries of doubly stochastic graph matrices and characterize all extremal graphs which attain the upper bound. In Section 3, we investigate the relations between distances for graph vertices and vertex degrees and present a proof of Theorem AN UPPER BOUND FOR THE MAIN DIAGONAL ENTRIES OF X(G) We need more notation. Let G be a simple graph on n vertices with doubly stochastic graph matrix Ω(G)=(ω ij ), define ω(g)=max{ω ii, i n}. A vertex of G is called dominating if it is adjacent to every other vertex in G. The main result of this section is the following. Theorem 2.. Let G be a simple connected graph on n vertices with doubly stochastic graph matrix Ω(G)=(ω ij ).Then ω(g) 2n + 2n 2n 2n with equality if and only if G is a path on n vertices, where =( 5+)/2. We first prove some preliminary results. Lemma 2.2. Let G be a simple graph on n vertices {v,...,v n } with Ω(G)=(ω ij ). Then d i + ω ii 2 d i +2. Moreover, if G is connected, then for i n the right hand equality holds if and only if v i is a dominating vertex and the left hand equality holds if and only if n=. Proof. Let L(G)=(l ij ). By equating the (i,i)-entry on both sides of the equation (I n +L(G))Ω(G)=I n, we see that (l ii +)ω ii + n j=,j i l ij ω ij =. Since n j= ω ij =, we have (d i +2)ω ii =2 ω ij, () v j v i

5 08 JOURNAL OF GRAPH THEORY where v j v i means that v j is not adjacent to v i and v j v i. Hence ω ii 2 v j v i ω ij d i +2 2 d i +2. If G is connected, then ω ij >0 for i,j n. Hence the above equality holds if and only if v j v i ω ij =0, i.e. if and only if v i is a dominating vertex. Similarly, we have (d i +)ω ii =+ v j v i ω ij, (2) where v j v i means that v j is adjacent to v i. Hence with equality if and only if n=. ω ii d i + Lemma 2.3. Let P n be a path on n vertices with doubly stochastic matrix Ω(P n )=(ω ij ).Letω(P n )=max{ω ii, i n}. Then where =( 5+)/2. ω(p n )= 2n + 2n 2n 2n, Proof. Let P n be the path with vertices v,...,v n in order. Then d =d n =and d 2 = =d n =2. By Lemma 2.2, ω ii 2 for i=2,...,n. By symmetry, ω =ω nn. We next show that ω(p n )=ω by showing that ω > 2. By equating the entries in the first column of the equation (I n +L(P n ))Ω(P n )=I n, we obtain n equations 2ω ω 2 =, ω +3ω 2 ω 3 =0,..., ω n 2, +3ω n, ω n =0, ω n, +2ω n =0. One can easily check that ω = 2 3 when n=2, and ω = 5 8 when n=3. For n 4, by eliminating ω n,...,ω 2 recursively from the last n equations, we see that (2+ ( 3+ ) n 2 5 5) (2 3 5)( ) n ω = ( 3+ ) n 5 3 ( ) n = 2n + 2n 5 2n 2n > Hence ω(p n )=ω, as desired. We are now ready to prove Theorem 2.. Proof of Theorem 2.. We prove the assertion by induction on n. Ifn= or2, the assertion clearly holds. Assume that the assertion holds for n. By Theorem in [3], we observe that deleting an edge from G cannot decrease the main diagonal entries of doubly stochastic graph matrix Ω(G). Hence without loss of generality, we only consider that G isatreeonn vertices.

6 VERTEX DEGREES 09 If d i 2, then by Lemmas 2.2 and 2.3, ω ii (2/(d i +2)) 2 <ω(p n). If d i =, without loss of generality, we assume that d =ande={v,v 2 } E(G). Then F =G e has two components with one component F with only one vertex v and the other component F 2 on n vertices. Let x=e e 2, where e i is a vector with the only nonzero entry a intheith component. Then L(G)=L(F)+xx T. By the Sherman Morrison formality (see, e.g. [8, p 9]), we have Ω(G)=Ω(F) Ω(F)xxT Ω(F) +x T Ω(F)x. Let Ω(F 2 )=(ω ij ) for 2 i,j n. SinceF 2 is a simple graph on n vertices, by the induction hypothesis, ω 22 ω(p n ) with equality if and only if F 2 is a path on n vertices. Therefore, by a calculation, ω = 2+ω 22 2+ω(P n ) =ω(p n) with equality if and only if G is a path on n vertices. 3. MERRIS QUESTION In order to investigate Merris question, we first prove the following lemmas. Lemma 3.. Let T be a tree on vertex set V ={v,...,v n } with doubly stochastic matrix Ω(T)=(ω ij ).If{v i v j } E(T), then ω ij d i d j +d i +d j with equality if and only if n=2. Proof. Without loss of generality, we may assume that i=r, j=r+. Observe that F =T {v i v j } has two components, one component F on vertex set V ={v,...,v r } and the other component F 2 on vertex set V 2 ={v r+,...,v n }. Let x=e r e r+, where e i is a vector with the only nonzero entry a in the ith component. Let Ω(F )=(ω ij ), i,j r and Ω(F 2 )=(ω ij ), r+ i,j n. Then Hence Ω(T)=Ω(F) Ω(F)xxT Ω(F) +x T Ω(F)x. ω rr ω r+,r+ ω r,r+ = +ω. rr +ω r+,r+ By Lemma 2.2, and d r (F )=d r (T), d r+ (F 2 )=d r+ (T), we have ω rr d r (T) +, ω r+,r+ d r+ (T) +

7 0 JOURNAL OF GRAPH THEORY with both equalities if and only if n=2. Therefore, by Lemma 2.2 in [7], ω r,r+ d r d r+ + + d r d r+ = d r d r+ +d r +d r+ with equality if and only if n=2. Remark. When G is a tree, Lemma 3. strengthens Theorem.2, since d i +d j n implies d i d j +d i +d j (n 2 +4n)/4. Hence with the aid of the value of entry of Ω(G), we may judge whether two vertices are adjacent. Lemma 3.2. Let G be a simple graph on vertex set V ={v,...,v n } with doubly stochastic matrix Ω(G)=(ω ij ).Ifd k 2d i, then ω kk <ω ii. Proof. By Equations () and (2) in Lemma 2.2, ω ii ω kk = d k 2d i +(d k +2) v j v i ω ij +(d i +) v j v k ω kj. (3) (d k +2)(d i +) It follows from d k 2d i that ω kk <ω ii. We also need a lemma from [3]. Lemma 3.3 (Merris [3]). Let G be a graph on n vertices {v,...,v n } with doubly stochastic graph matrix Ω(G)=(ω ij ).Thenv k is a least remote vertex if and only if ω kk is a main diagonal minimum entry. Theorem 3.4. Let G be a simple graph on n vertices {v,...,v n }.Ifd k 2d i for all i k, then r(k)=r(g). Proof. By Lemma 3.2, ω kk is the only main diagonal minimum entry. It follows from Lemma 3.3 that v k is a least remote vertex. So r(k)=r(g). Theorem 3.5. Let T be a tree on p+q 3 +q 2 +q+ vertices as in Figure with doubly stochastic graph matrix Ω(T)=(ω ij ).If2(q+) 3 p>q+, thenr(p)>r(t). Proof. Let q L(K,q )=

8 VERTEX DEGREES FIGURE. Tree T. and q e T e T... e T e L(K,q )+e e T L(T )= e 0 L(K,q )+e e T , e L(K,q )+e e T where e is a vector of size q with the only nonzero entry a in the first component. It is easy to see that I +L(K,q )+e e T and I +L(T )+e e T are positive definite. Let A = I +L(K,q )+e e T =(c ij) andb =I +L(T )+e e T =(f ij). Without loss of generality, we may assume that 2I p e e T p q+2 e T... e T I +L(T)= 0 0 e B, e 0... B

9 2 JOURNAL OF GRAPH THEORY where e is a vector of all ones with p components. Clearly, the (,)-entry of (I +L(K,q )) is α=2/(q+2). By some calculations, it is easy to see that c = α +α = 2 q+4. Hence by some calculations, the (,)-entry of (I +L(T )) is β= q+4 q 2 +3q+4 and f = β +β = q+4 q 2 +4q+8. Therefore, by the Schur complement, it is easy to see that the cofactor of (p,p) entry in I +L(T) is equal to 2 p (det B ) q (q+2 qb )=2 p (det B ) q q3 +5q 2 +2q+6 q 2 +4q+8 and the cofactor of (p+,p+) entry in I +L(T) is equal to 2 p (det B ) q p+3 2. Hence ω pp ω p+,p+ = 2p (det B ) q ( q 3 +5q 2 +2q+6 det(i +L(T)) q 2 +4q+8 since 2(q+) 3 p>q+. Hence we finish our proof. p+3 ) >0, 2 Now we present a proof of Theorem.6. Proof of Theorem.6. By Theorem 3.5, for any p and q with 2(q+) 3 p q, we have d(v p )=p+>d(v i ) for all i p and r(p)>r(t). Hence (i) holds. (ii) follows from Theorem 3.4. Lemma 3.6. Let T be a tree on n vertices {v,...,v n } and doubly stochastic graph matrix Ω(T)=(ω ij ).Ifd k 2d i for i k, then r(k)=r(t). Proof. By Lemma 3., Hence ω ij d i v j v i v j v i d i d j +d i +d j di 2 +2d i (d i +2) ω ij 2d2 i d i v j v i di 2 +2d i +. d k d i +d k +d i + 2d i + 2d 2 i +2d i.

10 VERTEX DEGREES 3 By Equation (3) in Lemma 3.2, ω ii ω kk = (d k +2) v j v i ω ij +(d i +) v j v k ω kj (d k +2)(d i +) If follows from Lemma 3.3 that the assertion holds. >0. Remark. By Theorem 3.5, there exists a class of graphs such that Question.5 has a negative answer if d k d i for all i k and d k 2d j 3 for some j k. Moreover, we also show that Question.5 has a positive answer for any tree with d k 2d i by Lemma 3.6. Using MATLAB, we see many examples with d k 2d i 2 that satisfy r(k)=r(g). The author does not know any examples with such parameters that do not satisfy the required condition. We conclude this article with the following question. Question 3.7. Let G be a simple connected graph on n vertices {v,...,v n } with the doubly stochastic graph matrix Ω(G)=(ω ij ).Ifd k =max{d,d 2,...,d n } and d k 2d i 2 for all i k, does r(k)=r(g) hold? ACKNOWLEDGMENTS The author thank the anonymous referees for their kind comments and suggestions. REFERENCES [] A. Berman and X. D. Zhang, A note on the degree antiregular graphs, Linear and Multilinear Algebra 47 (2000), [2] P. Yu. Chebotarev and E. V. Shamis, On the proximity measure for graph vertices provided by the inverse Laplacian characteristic matrix. Abstracts of the Conference Linear Algebra and its Applications, 0 2 July 995, University of Manchester, Manchester, UK, 995, pp Available from: higham/laa95/abstracts.ps [3] P. Yu. Chebotarev and E. V. Shamis, Matrix-forest theorems, arxiv paper math.co/ , 995. Available from: [4] P. Yu. Chebotarev and E. V. Shamis, The matrix-forest theorem and measuring relations in small social groups, Automat Remote Control 58(9) (997), [5] P. Chebotarev and E. V. Shamis, The forest metrics for graph vertices, Electron Notes Discrete Math (2002), [6] P. Chebotarev, Spanning forests and the golden ratio, Discrete Appl Math 56 (2008), [7] V. E. Golender, V. V. Drboglav, and A. B. Rosenblit, Graph potentials method and its application for chemical information processing, J Chem Inf Comput Sci 2 (98), [8] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, London, 985.

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