# 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 *u*_{0} ∈ *V*(*G*) such that *G* − *u*_{0} is a unicyclic graph. Denote *Q*(*n, d*_{0}) = {*G: G* is a quasi-unicyclic graph of order *n* with *G* − *u*_{0} being a unicyclic graph and *d*_{G}(*u*_{0}) = *d*_{0}}. Let *A*(*G*) be the adjacency matrix of a graph *G*, and let *λ*_{1}(*G*), *λ*_{2}(*G*),…, *λ*_{n}(*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 *S*_{k} (*G*). Let *S* (*G*) = (*S*_{0}(*G*), *S*_{1}(*G*),…, *S*_{n−1}(*G*)) be the sequence of spectral moments of *G*. For two graphs *G*_{1}, *G*_{2}, we have *G*_{1} ≺_{S}*G*_{2} if for some *k*(*k* = 1,2,…, *n*−1), and we have *S*_{i}(*G*_{1}) = *S*_{i}(*G*_{2}) (*i* = 0,1, …, *k*−1) and *S*_{k}(*G*_{1}) < *S*_{k}(*G*_{2}). In this paper, we determine the second to the fourth largest quasi-unicyclic graphs, in an *S*-order, in the set *Q*(*n, d*_{0}), respectively.

## Key words

spectral moment unicyclic graph quasi-unicyclic graph## CLC number

O 157.5## Preview

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