Eigenvalues and toughness of regular graphs

Abstract

The toughness of a graph G, denoted by $t\left(G\right)$, is defined as $t\left(G\right)=\min \{\frac{\left|S\right|}{c(G-S)}:S\subset V(G\left)\text{and}c\right(G-S)>1\}$. The bipartite toughness $\tau \left(G\right)$ of a non-complete bipartite graph $G=(X,Y)$ is defined as $\tau \left(G\right)=\min \{\frac{\left|S\right|}{c(G-S)}:S\subset X\text{or}S\subset Y,\text{and}c(G-S)>1\}$. Incorporating the toughness and eigenvalues of a graph, we provide two sufficient eigenvalue conditions for a regular graph to be $\frac{1}{b}-$tough for a positive integer b, which extend a significant result by Cioabă and Wong [10]. For a regular bipartite graph, it is proved that $\tau \left(G\right)\ge 1$. We further show a sufficient eigenvalue condition with the second largest eigenvalue for a regular bipartite graph having bipartite toughness more than 1.