Sufficient Conditions for Maximally k-Isoperimetric Edge Connectivity of Graphs

Abstract

The k-isoperimetric edge connectivity is a more refined network reliability index than edge connectivity. The k-isoperimetric edge connectivity of a connected graph G is defined as γk(G) = min{|[X, [Formula: see text]]| : X ⊆ V (G), |X| ≥ k, |[Formula: see text]| ≥ k}. Let βk(G) = min{|[X,[Formula: see text]]| : X ⊆ V (G), |X| = k}. A graph G is called a γk-optimal graph if γk(G) = βk(G). An edge cut S = [X,[Formula: see text]] is called a γk-cut if |S| = γk(G), X ⊆ V (G), |X| ≥ k and |[Formula: see text]| ≥ k. Moreover, G is called a super-γk graph if every γk-cut [X,[Formula: see text]] of G has the property that either |X| = k or |[Formula: see text]| = k. Let G be a graph of order at least 2k with k ≥ 2. In this paper, we prove that for any pair u, υ of nonadjacent vertices in G, if |N(u)∩N(υ)| ≥ k+1 when neither u nor υ lies on a triangle, or |N(u)∩N(υ)| ≥ 2k -1 when u or υ lies on a triangle, then G is γk-optimal. Moreover, if G is a triangle-free graph, and |N(u)∩υ(υ)| ≥ k +1 for all pairs u, υ of nonadjacent vertices in G, then G is either super-γk or isomorphic to Kk+1,k+1.