Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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 Σv ∈ 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). ,  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,  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 ll characterized graphs.
['Hong-Jian Lai', 'Xiangjuan Yao']
I am writing a research paper on a idea Group connectivity of graphs with diameter at most 2
[{"from": "human", "value": "I am writing a research paper on a idea Group connectivity of graphs with diameter at most 2"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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) \u21a6A satisfying \u03a3v \u2208 V(G) b(v) = 0, there is a function f: E(G)\u21a6A* such that at each vertex v \u2208 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). ,  investigate, for a 2-edge-connected graph G with diameter at most 2, the group connectivity number \u039b g(G) = min{n: G is A-connected for every abelian group A with |A| \u2265 n}, and show that any such graph G satisfies \u039b g (G) \u2264 6. Furthermore,  show that if G is such a 2-edge-connected diameter 2 graph, then \u039b g(G) = 6 if and only if G is the 5-cycle; and when G is not the 5-cycle, then \u039b g (G) = 5 if and only if G is the Petersen graph or G belongs to two infinite families of ll characterized graphs."}]