Open problem on $σ$-invariant

Abstract

Let $G$ be a graph of order $n$ with $m$ edges. Also let $μ_1\geq μ_2\geq \cdots\geq μ_{n-1}\geq μ_n=0$ be the Laplacian eigenvalues of graph $G$ and let $σ=σ(G)$ $(1\leq σ\leq n)$ be the largest positive integer such that $μ_σ\geq \frac{2m}{n}$. In this paper, we prove that $μ_2(G)\geq \frac{2m}{n}$ 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 $σ=1$.