Group connectivity of graphs with diameter at most 2

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