Spectral conditions for edge connectivity and spanning tree packing number in (multi-)graphs

A multigraph is a graph with possible multiple edges, but no loops. Let t be a positive integer. Let G_t be the set of simple graphs (or multigraphs) such that for each G∈G_t there exist at least t+1 non-empty disjoint proper subsets V₁,V₂,…,V_t+1⊆V(G) satisfying V(G)∖(V₁∪V₂∪⋯∪V_t+1)≠ϕ and edge connectivity κ^′(G)=e(V_i,V(G)∖V_i) for i=1,2,…,t+1. Let D(G) and A(G) denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) G, and let μ_i(G) be the ith largest eigenvalue of the Laplacian matrix L(G)=D(G)+A(G). In this paper, we investigate the relationship between μ_n−2(G) and edge connectivity or spanning tree packing number of a graph G∈G₁, respectively. We also give the relationship between μ_n−3(G) and edge connectivity or spanning tree packing number of a graph G∈G₂, respectively. Moreover, we generalize all the results about L(G) to a more general matrix aD(G)+A(G), where a is a real number with a≥−1.