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 is an element of G(t) there exist at least t + 1 non-empty disjoint proper subsets V-1, V-2, ... , Vt+1 subset of V(G) satisfying V(G) \ (V-1 boolean OR V-2 boolean OR ... boolean OR Vt+1) not equal phi and edge connectivity kappa'(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 mu(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 mu(n-2)(G) and edge connectivity or spanning tree packing number of a graph G is an element of G(1), respectively. We also give the relationship between mu(n-3)(G) and edge connectivity or spanning tree packing number of a graph G is an element of G(2), 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. (c) 2023 Elsevier Inc. All rights reserved.