Edge connectivity, packing spanning trees, and eigenvalues of graphs

Let (Formula presented.) be the set of simple graphs (or multigraphs) G such that for each (Formula presented.) there exists at least two non-empty disjoint proper subsets (Formula presented.) satisfying (Formula presented.) and edge connectivity (Formula presented.) for (Formula presented.). A multigraph is a graph with possible multiple edges, but no loops. Let (Formula presented.) 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 (Formula presented.), we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bounds of (Formula presented.) and (Formula presented.) of a simple graph or a multigraph (Formula presented.), respectively.