2 trees T 1,T 2,...,T

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1 Volume 3, pp , March 0 ORDERING TREES BY THE MINIMAL ENTRIES OF THEIR DOUBLY STOCHASTIC GRAPH MATRICES SHUCHAO LI AND QIN ZHAO Abstract Gien an n-ertex graph G, the matrix Ω(G) (I n + L(G)) (ω ij ) is called the doubly stochastic graph matrix of G, where I n is the n n identity matrix and L(G) is the Laplacian matrix of G Let ω(g) be the smallest element of Ω(G) Zhang and Wu [XD Zhang and JX Wu Doubly stochastic matrices of trees Appl Math Lett, 8: , 005] determined the tree T with the minimum ω(t) among all the n-ertex trees In this paper, as a continuance of the Zhang and Wu s wor, we determine the first trees T,T,,T such that ω(t ) < ( ) ( ) ω(t ) < < ω T ω T < ω(t), where T i is obtained by attaching a pendant ertex to i of path P i and T is an n-ertex tree different from the trees T,T,,T Key words Tree, Doubly stochastic graph matrix, Pendant ertex AMS subject classifications 05C50, 05C05 Introduction Let G (V G,E G ) be a simple graph with ertex set V G and the edge set E G Let deg G ( i ) or d i be the degree of ertex i and D(G) diag(d,d,,d n ) Let A(G) be the n n adjacency matrix whose (i,j)-entry is if i j E G and 0 otherwise The matrix L(G) D(G) A(G) is called the Laplacian matrix of G, which may be dated bac to Kirchhoff s Matrix-Tree Theorem and has been extensiely studied for the past fifty years (eg, see [5, 9] and the references therein) It is well nown that L(G) is positie semidefinite and singular Thus, its eigenalues can be arranged as λ λ λ λ n 0 The second smallest eigenalue λ, also denoted by α(g), is nown as the algebraic connectiity of G; see [0, 5] It follows from Kirchhoff s Matrix-Tree Theorem that λ > 0 if and only if G is connected Since the algebraic connectiity is releant to important problems regarding the diameter of graphs, the expanding properties of graphs, the combinatorial optimization problems, and the theory of elasticity, etc (for example, see [8, 9]), it has receied much more attention Recently, there is an excellent surey on algebraic connectiity of graphs written by de Abreu [], which is referred to the reader for further information Receied by the editors on October 3, 0 Accepted for publication on March 0, 0 Handling Editor: Michael Tsatsomeros Faculty of Mathematics and Statistics, Central China Normal Uniersity, Wuhan , PR China (lscmath@mailccnueducn) 95

2 Volume 3, pp , March SC Li and Q Zhao In the study of chemical information processing, Golender et al [] introduced another important matrix: doubly stochastic graph matrix associated with a graph, which may be used to describe some properties of the topological structure of chemical molecules Let I n be the n n identity matrix and Ω(G) (I n +L(G)) (ω ij ) It can be erified that Ω(G) is a doubly stochastic matrix (see, for example, []) Thus, Ω(G) is called the doubly stochastic graph matrix In [0] Pereira and Vali studied the spectra of doubly stochastic matrices On the other hand, Chebotare and Shamis [6] and Chebotare [5] pointed out that the doubly stochastic graph matrix may be used to measure the proximity among ertices and ealuate the group cohesion in the construction of sociometric indices and represent a random wal In particular, the diagonal entries ω ii of Ω(G) measure the peripherality (solitariness) of i [6] An important result on the relationship between the entry of the doubly stochastic matrix of a graph and its number of spanning forests, which is called the matrix forest theorem (995) (see Refs [7, 8]), is due to Chebotare and Shamis Shamis presented the result at the 995 International Linear Algebra Society (ILAS) Conference in Atlanta (see Ref [9]), and a proof of an earlier ersion of this theorem can be found in [] Moreoer, Merris [7] established the relationship between the entry ω ij of Ω(G) and structure of the simple n-ertex graph G: If ω ij < 4/(n +4n), then i is not adjacent to j Moreoer, ω ij is also relatie to the weights of routes of arious lengths between i and j (see [6], Proposition 0) Zhang [3] determined a sharp lower bound for the smallest entries, among those corresponding to edges, of doubly stochastic matrices of trees Zhang [4] also presented some relations between the diameter of a tree and the smallest entry ω(g) min{ω ij : i,j n} Recently, Zhang [5] studied the relationship between ertex degrees and entries of the doubly stochastic graph matrix Zhang and Wu[] obtained sharp upper and lower bounds for the smallest entries of doubly stochastic matrices of trees and characterized all extreme graphs which attain the bounds Motiated by these related results, we inestigate the property of the smallest entries of doubly stochastic matrices of graphs We identify the first n-ertex trees according to their smallest entries in the corresponding doubly stochastic matrices: ( ) ( ) ω(t ) < ω(t ) < < ω T ω T < ω(t), where T i is obtained by attaching a pendant ertex to i of path P i and T is an n-ertex tree different from the trees T,T,,T Preliminaries We define ω(g) max{ω ii : i n} A dominating ertex of G is a ertex of degree, ie, a ertex adjacent to eery other ertex Wecall apendant ertex, ifdeg G () Forconenience,letPV(G)bethesetofall pendant ertices of G, and we set T n, {T : T is an n-ertex tree with pendant

3 Volume 3, pp , March 0 Ordering Trees by the Minimum Entry of Their Doubly Stochastic Graph Matrices 97 ertices} The distance between ertices u and in G is denoted by d G (u,) Let N G () {w V G w E G } Clearly, deg G () N G () We first gie some lemmas that will be used in the proofs of our main results Lemma ([7]) Let G be a graph with doubly stochastic graph matrix Ω(G) (ω ij ) If r s, then ω rr ω rs, with equality if r is a dominating ertex of the connected component of G that contains s, or if deg G ( s ) and r s E G Lemma ([5]) Let G be a simple connected graph on n ertices with doubly stochastic graph matrix Ω(G) (ω ij ) Then ω(g) φ +φ n φ n φ n with equality if and only if G is a path on n ertices, where φ 5+ Lemma 3 Let P n be a path on n ertices with doubly stochastic matrix Ω(P n ) (ω ij ) Then ω > ω > > ω n n ω n n, the last equality holds if and only if n is een Proof This result immediately follows from Theorem in [5] and the main theorem in [] Lemma 4 ([4]) Let T be a tree of order n with ertex set V T {,,, n } If ω(t) min{ω ij : i,j n} ω l, then the following hold: (i) and l are two pendant ertices (ii) If the diameter d of T is no more than 4, then d G (, l ) d Lemma 5 ([3]) Let T be a tree of order n 4 with at least three pendant ertices, say,,, n Suppose that 3 N T ( ) and T T 3 + Let Ω(T) (ω ij ) and Ω(T ) (ω ij ) Then ω n > ω n Corollary 6 Let T T n, be a tree such that ω(t) is as small as possible, where n 4 and 3 are fixed Suppose that ω(t) ω n, is a pendant ertex of T different from, n and 3 N T ( ), then we hae deg T ( 3 ) > Proof Suppose to the contrary that deg T ( 3 ) Since ω(t) ω n by Lemma 4(i), we hae that, n PV(T) Let T T 3 + Then T T n, with doubly stochastic graph matrix Ω(T ) (ω ij ) By Lemma 5,

4 Volume 3, pp , March SC Li and Q Zhao we obtain A contradiction to the choice of T ω(t ) ω n < ω n ω(t) Corollary 7 For any tree T T n,+, there exists a tree T T n, such that ω(t) > ω(t ), where n 4, Proof Choose T T n,+ such that ω(t) is as small as possible Assume that ω(t) ω n, then, n PV(T) Note that Let be a pendant ertex of T different from, n and 3 N T ( ) Then, by Corollary 6, we hae deg T ( 3 ) > Let T T 3 + Then T T n, with doubly stochastic graph matrix Ω(T ) (ω ij ) By Lemma 5, we obtain ω(t ) ω n < ω n ω(t) Thus, for any tree T T n,+, there exists a tree T T n, such that ω(t) > ω(t ), as desired 3 Main results Let G be a simple graph obtained from graph G by attaching pendant ertices to a ertex of G ; see Fig 3 Then we characterize the ω(g) by the entries of Ω(G ) G + G Fig 3 Graph G Theorem 3 Let G be a graph of order n, which has pendant ertices,, attached to the same ertex + of G (see Fig 3) Let Ω(G) (ω ij ) for i,j n and Ω(G ) (ωij ) for + i,j n Then the smallest entry of Ω(G) satisfies { } ω(g) min ωij ω i,+ ω +,j +ω+,+, ω +,j +ω +,+

5 Volume 3, pp , March 0 Ordering Trees by the Minimum Entry of Their Doubly Stochastic Graph Matrices 99 Proof Let {}}{ x 0 0 T Then Ω(G) (I n +L(G)) (F +xx T ) with ( ) K 0 F 0 I n +L(G, ) where Hence, we hae F ( ) K 0 0 Ω(G ) K By the Sherman Morrison formula (see [3, p 9]), we hae Ω(G) F F xx T F +x T F x Note that +x T F x +ω +,+ and 0 ω+,n ωn,+ ωnn ω +,+ {}}{ F xx T F ω +,+ ω n,+ {}}{ ω +,+ ω n,+ T

6 Volume 3, pp , March SC Li and Q Zhao Hence, we hae and (+ω+,+ ), if i j ; ω ij ωij ω i,+ ω +,j +ω, if i j +; +,+ ω +,j, if i < j +ω +,+ + ω ii (+ω+,+ ), if i ; ωii ω i,+, if i + +ω +,+ (3) (3) By Lemma, it is straightforward to chec that for, and hence { ω(g) min ωij ω i,+ ω +,j +ω+,+ ; (+ω +,+ ) > ω +,j +ω +,+ } ω +,j +ω +,+ This completes the proof Corollary 3 Let G be a simple connected graph on n ertices with doubly stochastic graph matrix Ω(G) (ω ij ) If is a pendant ertex of G and i E G, then ω ii ω(p ) +ω(p ) with equality if and only if G is a path on n ertices Proof Let G G i with doubly stochastic graph matrix Ω(G ) (ω ij ) for i,j n By Lemma, we hae ω ii ω(p ) with equality if and only if G is a path on ertices and i is a pendant ertex of G Therefore, by Eq (3) in Theorem 3, we hae ω ii ω ii ω ii +ω ii ω + ii ω(p ) + ω(p ) +ω(p ) with equality if and only if G is a path on ertices and i is a pendant ertex of G, which is equialent to that G is a path on n ertices Corollary 33 For any P i, we hae (i) ω(p i+ ) ω(pi) +ω(p and ω(p i) i+) +ω(pi) +ω(p i), where i,,, (ii) ω (P i )+ω(p i ) ω (P i ) 0, where i,,,

7 Volume 3, pp , March 0 Ordering Trees by the Minimum Entry of Their Doubly Stochastic Graph Matrices 30 Proof By Theorem in [5], ω(p i ) Φ i and ω(p i ) Φi Φ i, where Φ are the Fibonacci numbers This yields (i) and, using Cassini s identity, (ii) Let T i (see Fig 3) be the tree on n ertices obtained from a path P i by attaching a new pendant ertex n at i n i n T i Fig 3 Graph T i ( Theorem 34 Let T i i,,, ) be a tree of order n, which is depicted in Fig 3, then we hae ( ) ( ) ω(t ) < ω(t ) < < ω T ω T with equality if and only if n is odd Proof Let x i be an n dimensional ector whose only nonzero component is in the ith position and x x n x i Thus, L(T i ) L(F)+xx T Let Ω(T i ) (ω ij ) and Ω(P ) (ω ij ) By the Sherman-Morrison formula (see, eg [3, p 9]), we hae Ω(T i ) Ω(F) Ω(F)xxT Ω(F) +x T Ω(F)x By Lemma 4(i), we hae ω(t i ) min{ω,,ω n,ω,n } Note that Hence, we obtain ω n ω iω in ω ii ω,n ω,iω in ω ii ω(t i ) ω, ω, ω i ω i, +ω ii ω i > ω iω i, ω ii ω,, ω,i > ω iω i, ω ii ω, ω, ω,ω ii +ω ii ( By Lemma 3, we hae ω(t ) < ω(t ) < < ω T ω, +ω ii ) ( ω T equality if and only if is een, which is equialent to that n is odd ω(p ) +ω ii ) with Theorem 35 Let T T n,4 and T i T n,3 where T i ( i,,, ) is depicted in Fig 3, then we hae ω(t i ) < ω(t)

8 Volume 3, pp , March SC Li and Q Zhao Proof Let T T n,4 with doubly stochastic graph matrix Ω(T) (ω ij ), such that ω(t) is as small as possible By Corollary 6, we obtain that T {ˆT, ˆT, ˆT 3 }; see Fig 33 In what follows, if two graphs, say G and H, are isomorphic graphs, then we write G H n n i n ˆT i j n ˆT n n n n n l+ n s s+ l ˆT 3 T T Fig 33 Graphs ˆT, ˆT, ˆT 3,T and T If T ˆT, then T i i n Pn Let Ω(P n ) (ω pq) By Theorem 3, [4, Theorem 3], and Lemma 3, we hae ω(t) ω,n ω,n ω i ω i,n +ω ii ω,n ω ii ω,n +ω ii ω,n +ω ii ω,n +ω ω(t ) If T ˆT, then T j n Ti Let Ω(T i ) (ω ij ) By Theorem 3 and [4, Theorem 3], we hae ω(t) ω,n ω,n ω i ω i,n +ω ii ω,n ω ii ω,n +ω ii ω,n +ω ii Since ω ii ω ii ω ij +ω, ω jj,n ω,n +ω and ω jj ij ω ii ω jj ω,n ω, together with i ω j,n Lemma, [4, Theorem 3], and Lemma 3, we obtain ω(t) 4ω,n 4+ω ii +ω jj +ω ii ω jj ω ij 4ω,n 4+ω ii +ω jj +ω ii ω jj ( ω ii ω jj ω,n ω i ω j,n ) 4ω,n 4+ω ii +ω jj +ω ii ω jj ( 4ω i ω j,n ω,n ω i ω j,n 4ω,n 4+ω ii +ω jj +ω ii ω jj 6ω,n 4ω,n 4+ω +ω n 3,n 3 +ω ω n 3,n 3 6ω,n ) ω(t )

9 Volume 3, pp , March 0 Ordering Trees by the Minimum Entry of Their Doubly Stochastic Graph Matrices 303 If T ˆT 3, let Ts T i i n and Ω(Ts) (ωpq) Then Ts T n,3 and ω(p n ) < ωpq for p,q n By Theorem 3, [4, Theorem 3], Lemma, and Corollary 3, we hae ω(t) ω l ω l ω,n ω n,l +ω n,n ω l ω l [ ω s ω s,n ω ss > ω(p n ) +ω n,n ω s,n ω sl ω ss +ω n,n ω s,n (+ω n,n ) ω l ] ω(p n ) [ > ω l ω s,n (+ωn,n )ω ss ( ) ω n,n +ω n,n ] ω l +ω n,n For i, by Theorem 3 and Corollary 33, we get ω(t i ) ω, +ω ii ω(p ) +ω ii +ω ii ω(p n ) +ω(p n ) < ω(p n ) Therefore, ω(t i ) < ω(t) for any tree T ˆT 3 Furthermore, note that ω(t i ) +ω ii +ω ii ω(p n ) +ω(p n ) +ω ii ω(p n 4 ) 3+8ω(P n 4 ), ω(p n 3 ) 5+3ω(P n 3 ) ω(t ) ω,n +3ω ω(p n 3) +3ω(P n 3 ) ω(p n 4) 7+5ω(P n 4 ), ω(t ) ω,n +ω ω(p n 4) +ω(p n 4) [ + ω(p n 4 ) ω (P n 4) +ω(p n 4) ω(p n 4 ) 4[ω(P n 4 )+ω(p n 4 )+][ω(p n 4 ) ω(p n 4 )+] ω(p n 4) 8+4ω(P n 4 ), and hence it is easy to obtain ω(t i ) < ω(t ) < ω(t ) for ω(p n 4) < ω(p n 4 ) < Thus, ω(t i ) < ω(t), where T T n,4 and i,,, Our last main result of this paper is to generalize [, Theorem ] We identify the first trees on n ertices according to their smallest entries in the corresponding doubly stochastic matrices Theorem 36 Let T be a tree of order n If T T i, where T i (i,,, ]

10 Volume 3, pp , March SC Li and Q Zhao ) is depicted in Fig 3, then we hae ( ) ( ) ω(t ) < ω(t ) < < ω T ω T < ω(t) with equality if and only n is odd Proof Theorem 36 follows directly from Corollary 7, Theorem 34, and Theorem 35 This completes the proof Acnowledgment This wor is financially supported by the National Natural Science Foundation of China (grant no 07096) and the Special Fund for Basic Scientific Research of Central Colleges (CCNUA005) The authors would lie to express their sincere gratitude to the referees for their careful reading of the paper and for all their insightful comments and aluable suggestions, which led to a number of improements in this paper REFERENCES [] NMM de Abreu Old and new results on algebraic connectiity of graphs Linear Algebra Appl, 43:53 73, 007 [] KT Atanasso, R Knott, K Ozei, AG Shannon, and L Szalay Inequalities among related pairs of Fibonacci numbers Fibonacci Quart, 4():0, 003 [3] A Berman and RJ Plemmons Nonnegatie Matrices in the Mathematical Sciences Academic Press, SIAM, 994 [4] A Berman, XD Zhang A note on the degree antiregular graphs Linear Multilinear Algebra, 47:307 3, 000 [5] PY Chebotare Spanning forests and the golden ratio Discrete Appl Math, 56(5):83 8, 008 [6] PY Chebotare and E V Shamis A martix-forest theorem and the measurement of relations in small social groups Automat Remote Control, 58(9):505 54, 997 [7] PY Chebotare and EV Shamis On the proximity measure for graph ertices proided by the inerse Laplacian characteristic matrix Abstracts of the Conference Linear Algebra and its Applications, Uniersity of Manchester, 6 7, 0 July, 995 Aailable at higham/laa95/abstractsps [8] PY Chebotare and EV Shamis Matrix-forest theorems Preprint, 995 Aailable at [9] PY Chebotare and EV Shamis On the proximity measure for graph ertices proided by the inerse Laplacian characteristic matrix Fifth Conference of the International Linear Algebra Society, 30 3, 995 [0] M Fiedler Algebraic connectiity of graphs Czechosloa Math J, 3(98):98 305, 973 [] VE Golender, VV Drbogla, and AB Rosenblit Graph potentails method and its application for chemical information processing J Chem Inf Comput Sci, :96 04, 98 [] R Grone and R Merris Ordering trees by algebraic connectiity Graphs Combin, 6:9 37, 990 [3] RA Horn and CR Johnson Matrix Analysis Cambridge Uniersity Press, London, 985 [4] JJ McDonald, M Neumann, H Schneider, and MJ Tsatsomeros Inerses of unipathic M- matrices SIAM J Matrix Anal Appl, 7:05 036, 996

11 Volume 3, pp , March 0 Ordering Trees by the Minimum Entry of Their Doubly Stochastic Graph Matrices 305 [5] R Merris Laplacian matrices of graphs: a surey Linear Algebra Appl, 97/98:43 76, 994 [6] R Merris Doubly stochastic graph matrices Uni Beograd Publ Eletrotehn Fa Ser Mat, 8:64 7, 997 [7] R Merris Doubly stochastic graph matrices II Linear Multilinear Algebra, 45:75 85, 998 [8] B Mohar Eigenalues, diameter, and mean distance in graphs Graphs Combin, 7:53 64, 99 [9] B Mohar Some applications of Laplace eigenalues of graphs In G Hahn and G Sabidussi (editors), Graph Symmetry, Kluwer Academic Publishers, Dordrecht, 5 75, 997 [0] R Pereira and MA Vali Inequalities for the spectra of symmetric doubly stochastic matrices Linear Algebra Appl, 49: , 006 [] E Shamis Graph-theoretic interpretation of the generalized row sum method, Math Social Sci, 7:3 333, 994 [] XD Zhang and JX Wu Doubly stochastic matrices of trees Appl Math Lett, 8: , 005 [3] XD Zhang A note on doubly stochastic graph matrices Linear Algebra Appl, 407:96 00, 005 [4] XD Zhang Algebraic connectiity and doubly stochastic tree matrices Linear Agebra Appl, 430: , 009 [5] XD Zhang Vertex degrees and doubly stochastic graph matrices J Graph Theory, 66:04 4, 0