Edge connectivity, packing spanning trees, and eigenvalues of graphs

Let G be the set of simple graphs (or multigraphs) G such that for each G∈G there exists at least two non-empty disjoint proper subsets V1,V2⊆V(G) satisfying V(G)∖(V1∪V2)≠φ and edge connectivity κ′(G)=e(Vi,V(G)∖Vi) for 1≤i≤2. A multigraph is a graph with possible multiple edges, but no loops. Let τ(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 τ(G), we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of κ′(G) and τ(G) of a simple graph or a multigraph G∈G, respectively.