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