Edge Connectivity, Packing Spanning Trees, and Eigenvalues of Graphs

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq \phi$ and edge connectivity $\kappa'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $\tau(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $\tau(G)$, we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bound of $\kappa'(G)$ and $\tau(G)$ of a simple graph or a multigraph $G\in\mathcal{G}$, respectively.