2 problems

Co-diameter-two characterization conjecture. The graph $G$ is non-trivially minimally tough if and only if $G$ is isomorphic to $S_{\ell,\ell,\ell}$ for some $\ell\geq 1$.

Let $G$ be a simple graph with minimum degree $\delta$. Let $\mu_2$ and $\mu_n$ denote the second-smallest and largest eigenvalues of the Laplacian matrix of $G$, respectively, and…