ON MAXIMAL ENERGY AND HOSOYA INDEX OF TREES WITHOUT PERFECT MATCHING

摘要

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