A multigraph is a graph that may have multiple edges, but has no loops. The multiplicity of a multigraph is the maximum number of edges between any pair of vertices. The spanning tree packing number of a graph G, denoted by (G), is the maximum number of edge-disjoint spanning trees contained in G. 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. By b (G), e (G), and (G), we denote the number of balloons, the size, and the vertex-connectivity of G, respectively. In this paper, we show that for a positive integer k and any multigraph G of order n 2r with multiplicity m k and minimum degree 2k, if e (G) mr2+n-r2+k, then (G) k, where r= (+1) /m. This extends the result of Fan, Gu and Lin (J. Graph Theory, 2023). Analogous results involving the size to characterize (G) k or b (G) k-1 of a multigraph G are also presented. In addition, we prove a tight sufficient condition to guarantee b (G) k-1 in terms of the spectral radius of a simple graph G, with extremal graphs characterized.
Cheng et al. (Mon,) studied this question.