Open problem on $\sigma$-invariant

TL;DR

This paper investigates the properties of Laplacian eigenvalues related to the $\sigma$-invariant of graphs, proving a bound for almost all graphs, characterizing extremal cases, and solving an open problem about graphs with $\sigma=1$.

Contribution

It establishes a lower bound for the second Laplacian eigenvalue for almost all graphs, characterizes extremal graphs, and solves an open problem regarding graphs with $\sigma=1$.

Findings

Proves $(G)\u2265m/n$ for almost all graphs.

Characterizes extremal graphs for the $\sigma$-invariant.

Provides a complete characterization of graphs with $\sigma=1$.

Abstract

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq \cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer such that $\mu_{\sigma}\geq \frac{2m}{n}$. In this paper, we prove that $\mu_2(G)\geq \frac{2m}{n}$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in \cite{KMT}, that is, the characterization of all graphs with $\sigma=1$.