Characterizations of minimal graphs with equal edge connectivity and spanning tree packing number

With graphs considered as natural models for many network design problems, edge connectivity ${\kappa }^{\mathrm{\prime }}(G)$ and maximum number of edge-disjoint spanning trees $\tau (G)$ of a graph $G$ have been used as measures for reliability and strength in communication networks modeled as graph $G$ (see \cite{Cunn85, Matula87}, among others). Mader \cite{Mader71} and Matula \cite{Matula72} introduced the maximum subgraph edge connectivity $\overline{\kappa’}(G)=\max {\kappa’(H): H \mbox{ is a subgraph of } G }$. Motivated by their applications in network design and by the established inequalities $$\overline{{\kappa }^{\mathrm{\prime }}}(G)\ge {\kappa }^{\mathrm{\prime }}(G)\ge \tau (G),$$ we present the following in this paper: (i) For each integer $k>0$, a characterization for graphs $G$ with the property that $\overline{{\kappa }^{\mathrm{\prime }}}(G)\le k$ but for any edge $e$ not in $G$, $\overline{{\kappa }^{\mathrm{\prime }}}(G+e)\ge k+1$. (ii) For any integer $n>0$, a characterization for graphs $G$ with $|V(G)| = n$suchthat$\kappa’(G) = \tau(G)$with$|E(G)|$ minimized.