On the Spectral Moment of Quasi-Unicyclic Graphs

Abstract

A connected graph G = (V(G), E(G)) is called a quasi-unicyclic graph, if there exists u0 ∈ V(G) such that G − u0 is a unicyclic graph. Denote Q(n, d0) = {G: G is a quasi-unicyclic graph of order n with G − u0 being a unicyclic graph and dG(u0) = d0}. Let A(G) be the adjacency matrix of a graph G, and let λ1(G), λ2(G),…, λn(G) be the eigenvalues in non-increasing order of A(G). The number $$\sum\limits_{i = 1}^n {\lambda _i^k(G)} $$ (k = 0,1, …, n−1) is called the k-th spectral moment of G, denoted by Sk (G). Let S (G) = (S0(G), S1(G),…, Sn−1(G)) be the sequence of spectral moments of G. For two graphs G1, G2, we have G1 ≺SG2 if for some k(k = 1,2,…, n−1), and we have Si(G1) = Si(G2) (i = 0,1, …, k−1) and Sk(G1) < Sk(G2). In this paper, we determine the second to the fourth largest quasi-unicyclic graphs, in an S-order, in the set Q(n, d0), respectively.