Sufficient conditions for super -restricted edge connectivity in graphs of diameter 2

For a connected graph G = ( V , E ) , an edge set S ⊆ E is a k -restricted edge cut if G − S is disconnected and every component of G − S has at least k vertices. The k -restricted edge connectivity of G , denoted by λ k ( G ) , is defined as the cardinality of a minimum k -restricted edge cut. Let ξ k ( G ) = min { | [ X , X ¯ ] | : | X | = k , G [ X ] is connected } . G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Moreover, G is super- λ k if every minimum k -restricted edge cut of G isolates one connected subgraph of order k . In this paper, we prove that if | N G ( u ) ∩ N G ( v ) | ≥ 2 k − 1 for all pairs u , v of nonadjacent vertices, then G is λ k -optimal; and if | N G ( u ) ∩ N G ( v ) | ≥ 2 k for all pairs u , v of nonadjacent vertices, then G is either super- λ k or in a special class of graphs. In addition, for k -isoperimetric edge connectivity, which is closely related with the concept of k -restricted edge connectivity, we show similar results.