Note on the Minimal Energy Ordering of Conjugated Trees

Abstract

AbstractFor 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(G). Gutman proposed two conjectures on the minimal energy of the class of conjugated trees (trees having a perfect matching). Zhang and Li determined the trees in the class with the minimal and second-minimal energies, which confirms the conjectures. Zhang and Li also found that the conjugated tree with the third-minimal energy is one of the two graphs which are quasi-order incomparable. Recently, Huo, Li and Shi found there exists a fixed positive integer N 0, such that for all n > N 0, the energy of the graphs with the third-minimal through the sixth-minimal are determined. In this paper, the N 0 is fixed by a recursive method, and the problem is solved completely.Keywordsextremal graphminimal energyquasi-order incomparableconjugated tree