Are there any attempts to characterize (simple, finite) graphs with vertex-connectivity $\kappa$ equal edge-connectivity $\kappa'$?
In general, $\kappa(G)\leq \kappa'(G)\leq \delta(G)$ where $\delta(G)$ denotes the minimum degree of $G$.
If characterizations are not known (quite possible), I would appreciate lists of widely known necessary or sufficient conditions. The following is a sufficient condition I found in the literature.
If $G$ is a connected edge-transitive graph, then $\kappa(G)=\kappa'(G)=\delta(G)$ (see Corollay 1A in [1]).
[1] Watkins, M. E., Connectivity of transitive graphs, J. Comb. Theory 8, 23-29 (1970). ZBL0185.51702.