ON MAXIMAL ENERGY AND HOSOYA INDEX OF TREES WITHOUT PERFECT MATCHING

E-ISSN： 1755-1633|81|1|47-57

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.81, Iss.1, 2010-02, pp. : 47-57

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

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.