On the spectral moment of graphs with $k$ cut edges

TL;DR

This paper investigates the spectral moments of graphs with a fixed number of cut edges, identifying extremal graphs in the spectral order within this class.

Contribution

It characterizes the extremal graphs in spectral order among connected graphs with a given number of cut edges, based on their spectral moments.

Findings

Identifies the first, second, last, and second last graphs in S-order within the class.

Provides a method to compare spectral moments for graphs with cut edges.

Enhances understanding of spectral properties related to graph connectivity and cut edges.

Abstract

Let $A(G)$ be the adjacency matrix of a graph $G$ with $\lambda_{1}(G)$, $\lambda_{2}(G)$, ..., $\lambda_{n}(G)$ being its eigenvalues in non-increasing order. Call the number $S_k(G):=\sum_{i=1}^{n}\lambda_{i}^k(G) (k=0,1,...,n-1)$ the $k$th spectral moment of $G$. Let $S(G)=(S_0(G),S_1(G),...,S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, we have $G_1\prec_sG_2$ if $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ for some $k\in {1,2,...,n-1}$. Denote by $\mathscr{G}_n^k$ the set of connected $n$-vertex graphs with $k$ cut edges. In this paper, we determine the first, the second, the last and the second last graphs, in an $S$-order, among $\mathscr{G}_n^k$, respectively.