Toughness and spectral radius in graphs

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