Edge-disjoint spanning trees and balloons in (multi-)graphs from size or spectral radius

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 $\tau(G)$, 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 $\kappa(G)$, 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 $n\geq 2r$ with multiplicity $m\leq k$ and minimum degree $\delta \geq2k$, if $e(G)\geq m[\binom{r}{2}+\binom{n-r}{2}]+k,$ then $\tau(G)\geq k$, where $r=\lceil(\delta+1)/m\rceil$. This extends the result of Fan, Gu and Lin (J. Graph Theory, 2023). Analogous results involving the size to characterize $\kappa(G)\geq k$ or $b(G)\leq k-1$ of a multigraph $G$ are also presented. In addition, we prove a tight sufficient condition to guarantee $b(G)\leq k-1$ in terms of the spectral radius of a simple graph $G$, with extremal graphs characterized.