Determining the conjugated trees with the third-through the sixth-minimal energies

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)521-532
Journal / PublicationMatch
Volume65
Issue number2
Publication statusPublished - 2011

Abstract

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. A conjugated tree is a tree that has a perfect matching. The conjugated trees Fn and B n with the minimal and secondminimal energies were determined by Zhang and Li. They also figured out that the conjugated trees with the third- and fourth-minimal energies are Ln or Mn. However, they could not determine which is the third, and the other is the fourth. Recently, S. Li and N. Li further investigated the conjugated trees with the third-, through the sixth-minimal energies. As a result, they figured out that these trees must be among the trees Ln, Mn, In and Wn/2 *. They then showed that the energy of M n is smaller than that of In, and the energy of L n is smaller than that of Wn/2 *, but they could not give a total ordering of the 4 trees. For comparing of the energies, a common used method is to compare the number of fc-matchings in each concerned tree, but it is often invalid for further comparing. This paper is aimed at solving the above unsolved problems by giving the energies of the 4 trees a total ordering, that is, completely determining the conjugated trees with the third-, forth-, fifth- and sixth-minimal energies. Our method uses the well-known Coulson integral formula.

Citation Format(s)

Determining the conjugated trees with the third-through the sixth-minimal energies. / Huo, Bofeng; Li, Xueliang; Shi, Yongtang et al.
In: Match, Vol. 65, No. 2, 2011, p. 521-532.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review