Toughness and spectral radius in graphs

Abstract

The Brouwer's toughness conjecture states that every $d$-regular connected graph always has $t(G)>\frac{d}{\hat{I}}»-1$ where $\hat{I}»$ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $\ddot{I}„(G)$ of a graph $G$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be $\ddot{I}„$-tough ($\ddot{I}„\ge 2$ is an integer) and to be $\ddot{I}„$-tough ($\frac{1}{\ddot{I}}„$ is a positive integer) with minimum degree $\hat{I}´$, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs.