On graphs with algebraic connectivity equal to minimum edge density

Given an undirected graph G=(V,E) and a nonempty subset X V, the edge density of X is given by ρ(X)=VEX/XV X, where EX is the set of all edges with one end in X and the other end in V X. It is known that the algebraic connectivity of G, denoted by a(G), satisfies a(G) minX Vρ(X). In this paper we study the graphs G for which equality holds in the above inequality.