Group connectivity of graphs with diameter at most 2 (Q820087)

Every undirected graph \(G\) is assigned a group connectivity number \(\Lambda_g(G):=\min\{k\mid\) if \(A\) is an abelian group with \(|A|\geq k\), then \(G\in\langle A\rangle\}\); \(\langle A\rangle\) denotes the family of graphs that are \(A\)-connected, where a graph \(G\) is said to be \(A\)-connected if \(G\) has an orientation such that for every \(b:V(G)\to A\) the graph \(G\) has an \((A,b)\)-nowhere-zero flow. The main results are the following. If \(G\) is a 2-edge-connected loopless graph with diameter at most 2, then \(\Lambda_g(G)\leq 6\), where \(\Lambda_g(G)=6\) if and only if \(G\) is the 5-cycle. When \(G\) is not the 5-cycle, then \(\Lambda_g (G)=5\) if and only if \(G\) is the Petersen graph or \(G\) belongs to two infinite families of well-characterized graphs.