Toughness and spectral radius in graphs

The toughness $t\left(G\right)$ of a non-complete graph G is defined as $t\left(G\right)=\min \left\{\frac{\left|S\right|}{c(G-S)}\right\}$ in which the minimum is taken over all proper sets $S\subset G$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. Conjectured by Brouwer and proved by Gu, a toughness theorem state that every d-regular connected graph always has $t\left(G\right)>\frac{d}{\lambda }-1$, where λ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $\tau \left(G\right)$ of a graph G, which is defined by $\tau \left(G\right)=\min \{\frac{\left|S\right|}{c(G-S)-1},S\subset V(G\left)\text{and}c\right(G-S)>1\}$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be τ-tough ($\tau \ge 2$ is an integer) and to be τ-tough ($\frac{1}{\tau }$ is a positive integer) with minimum degree δ, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs.