Group connectivity of graphs with diameter at most 2

Let G be an undirected graph, A be an (additive) abelian group and A ∗ = A − { 0 } . A graph G is A -connected if G has an orientation D ( G ) such that for every function b : V ( G ) ↦ A satisfying ∑ v ∈ V ( G ) b ( v ) = 0 , there is a function f : E ( G ) ↦ A ∗ such that at each vertex v ∈ V ( G ) , the amount of f values on the edges directed out from v minus the amount of f values on the edges directed into v equals b ( v ) . In this paper, we investigate, for a 2-edge-connected graph G with diameter at most 2, the group connectivity number Λ g ( G ) = min { n : G  is  A -connected for every abelian group  A  with  | A | ≥ n } , and show that any such graph G satisfies Λ g ( G ) ≤ 6 . Furthermore, we show that if G is such a 2-edge-connected diameter 2 graph, then Λ g ( G ) = 6 if and only if G is the 5-cycle; and when G is not the 5-cycle, then Λ g ( G ) = 5 if and only if G is the Petersen graph or G belongs to two infinite families of well characterized graphs.