Connections between Wiener index and matchings

Yan, Weigen; Yeh, Yeong‐Nan

doi:10.1007/s10910-005-9026-0

Cited by 37 publications

(12 citation statements)

References 29 publications

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“…If G is a tree, the coefficient c 2 is equal to its Wiener index, which is the sum of distances between all pairs of vertices [17]. In other words, we have Let η denote the nullity of the matrix Q(G).…”

Section: Elamentioning

confidence: 99%

On the log-concavity of Laplacian characteristic polynomials of graphs

Kiani¹,

Mirzakhah²

2014

ELA

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Abstract. Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients of G is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the algebraic connectivity of G. Finally, we investigate the mode of such sequences.

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Section: Elamentioning

confidence: 99%

On the log-concavity of Laplacian characteristic polynomials of graphs

Kiani¹,

Mirzakhah²

2014

ELA

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“…(The Wiener‐index of a graph is the sum

\sum_{x, y} d (x, y)

, where

d (x, y)

is the distance of the vertices x and y .) One can consider Theorem as a generalization of this fact since the signless coefficient of x 2 in the Laplacian polynomial is just the Wiener‐index (see ). Theorem (Second part.)

a (T^{'}) \geq a (T) .

…”

Section: The Laplacian Characteristic Polynomialmentioning

confidence: 99%

On a Poset of Trees II

Csíkvári

2012

J. Graph Theory

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In this article, we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the so-called generalized tree shift which seems to be a powerful tool to attack the problems of the above-mentioned kind. We show that the generalized tree shift increases the largest eigenvalue of the adjacency matrix and Laplacian matrix, decreases the coefficients of the characteristic polynomials of these matrices in absolute value. We will prove similar theorems for the independence polynomial and the edge cover polynomial. The generalized tree shift induces a partially ordered set on trees having fixed number of vertices. The smallest element of this poset is the path, largest element is the star. Hence, the above-mentioned results imply the extremality of the path and the star for these parameters.

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“…In particular, we have c 0 = 1, c n = 0, c 1 = 2|E(G)|, c n−1 = nτ (G), where τ (G) is the number of spanning trees of G (see [9]). If G is a tree, the Laplacian coefficient c n−2 is equal to its Wiener index, which is the sum of all distances between unordered pairs of vertices of G (see [1], [15]). In general, Laplacian coefficients c k can be expressed in terms of subtree structures of G.…”

Section: Introductionmentioning

confidence: 99%

Laplacian coefficients of unicyclic graphs with the number of leaves and girth

Zhang

2014

APPL ANAL DISCRETE M

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Motivated by Ilic and Ilic?s conjecture [A. Ilic, M. Ilic, Laplacian coefficients of trees with given number of leaves or vertices of degree two, Linear Algebra Appl., 431(2009)2195-2202.], we investigate properties of the minimal elements in the partial set (Ugn,l,?) of the Laplacian coefficients, where Ug n,l denote the set of n-vertex unicyclic graphs with the number of leaves l and girth g. These results are used to disprove their conjecture. Moreover, the graphs with minimum Laplacian-like energy in Ug n,l are also studied.

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Connections between Wiener index and matchings

Cited by 37 publications

References 29 publications

On the log-concavity of Laplacian characteristic polynomials of graphs

On the log-concavity of Laplacian characteristic polynomials of graphs

On a Poset of Trees II

Laplacian coefficients of unicyclic graphs with the number of leaves and girth

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