  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 ($\tau\geq 2$ 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.
