Toughness in regular graphs from eigenvalues

Abstract: The {\it toughness} $\tau(G)=\mathrm{min}{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G}$ for $G\ncong K_n,$ which was initially proposed by Chv\'{a}tal in 1973. A graph $G$ is called {\it $t$-tough} if $\tau(G)\geq t.$ Let $\lambda_i(G)$ be the $i$-th largest eigenvalue of the adjacency matrix of a graph $G$. In 1996, Brouwer conjectured that $\tau(G)\geq\frac{d}{\lambda}-1$ for a connected $d$-regular graph $G,$ where $\lambda=\mathrm{min}{|\lambda_2|, |\lambda_n|}.$ Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result $\tau(G)\geq\frac{d}{\lambda}-1,$ we know that if $G$ is a connected $d$-regular graph and $\lambda\leq\frac{bd}{b+1}$, then $\tau(G)\geq\frac{1}{b}$ for an integer $b\geq1.$ Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioab\u{a} et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if $G$ is a connected $d$-regular graph and $\lambda_2(G)<\phi(d,b)$, then $\tau(G)\geq\frac{1}{b}$. Meanwhile, we construct graphs implying that the upper bound on $\lambda_2(G)$ is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of $\lambda_{b+1}(G)$ to guarantee a connected $d$-regular graph to be $\frac{1}{b}$-tough.