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ON MAXIMAL ENERGY AND HOSOYA INDEX OF TREES WITHOUT PERFECT MATCHING

Part of: Graph theory

Published online by Cambridge University Press:  27 July 2009

HONGBO HUA
HONGBO HUA*
Affiliation:
Department of Computing Science, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China (email: hongbo.hua@gmail.com)
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Abstract

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Let G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

Keywords

treeperfect matchingenergy of graphspectra of graphHosoya indexk matchings

MSC classification

Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05C35: Extremal problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 47 - 57
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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