Spectral conditions for edge connectivity and packing spanning trees in multigraphs

A multigraph is a graph with possible multiple edges, but no loops. The multiplicity of a multigraph is the maximum number of edges between any pair of vertices. We prove that, for a multigraph G with multiplicity m and minimum degree δ≥2k, if the algebraic connectivity is greater than min⁡{2k−1⌈(δ+1)/m⌉,2k−12}, then G has at least k edge-disjoint spanning trees; for a multigraph G with multiplicity m and minimum degree δ≥k, if the algebraic connectivity is greater than min⁡{2(k−1)⌈(δ+1)/m⌉,k−1}, then the edge connectivity is at least k. These extend some earlier results.A balloon of a graph G is a maximal 2-edge-connected subgraph that is joined to the rest of G by exactly one cut-edge. We provide spectral conditions for the number of balloons in a multigraph, which also generalizes an earlier result.