Toughness and $Q$-spectral radius of graphs involving minimum degree

Abstract

In 1973, Chv$\mathrm{\acute{a}}$tal initially proposed the concept of toughness, which serves as a simple way to measure how tightly various pieces of a graph hold together. Let $G$ be a non-complete graph and let $t$ be a real number. If for every vertex cut set $S$ of $G$, $|S|\geq tc(G - S)$, then we say that $G$ is {\it $t$-tough}. The largest $t$ such that $G$ is $t$-tough is called the {\it toughness} of $G$ and is denoted by $t(G)$. Recently, Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented sufficient conditions based on the spectral radius for a graph to be 1-tough with minimum degree $\delta$ and $t$-tough with $t\geq 1$ being an integer, respectively. Inspired by their work, we in this paper consider the $Q$-spectral versions of the above two problems. Moreover, we also provide a sufficient condition in terms of the $Q$-spectral radius to for a graph to be $t$-tough with $\frac{1}{t}$ being a positive integer.