Group Connectivity Of Some Special Graphs

Graphs in this paper are finite and loops and multiple edges are permitted. Throughout the paper, Zn denotes the cyclic group of order n, for some integer n(?)2.Let A denote a nontrivial (additive) Abelian group with identity 0, and A*=A-{0}. Let F(G,A) denote the set of all functions from E(G) to A, and F*(G,A) denote the set of all functions from E(G) to A*. Unless otherwise stated, we shall adopt the following convention:if X(?)E(G) and if f:Xâ†'A is a function, then we regard f as a function f:E(G)â†'A such that f(e)=0 for all e∈E(G)-X.Given a function f∈F(G,A), let (?)f:V(G)â†'A be given byA function b:V(G)â†'A is called an A-valued zero-sum function on G if∑v∈V(G)b(v)=0 in G. The set of all A-valued zero-sum functions on G is denoted by Z(G,A). Given b∈Z(G,A) and an orientation D of G, a function f∈F*(G,A) is an (A,b)-nowhere-zero flow if (?)f=b. A graph G is A-connected if G has an orientation D such that for any b∈Z(G,A), G has an (A,b)-NZF. For an Abelian group A, let<A> be the family of graphs that are A-connected.For a 2-edge-connected graph G, the group connectivity of G is defined as:∧g(G)=min{k:G is A-connected for every Abelian group A with|A|(?)K) We show that if G is 2-edge-connected, then∧g(G) exists as a finite number.In this paper, by the connectivity of AG4, we find a counter-example to the fol-lowing Devos's conjecture, that is:'Let G be a 4-edge-connected graph with each edge contained in a circuit of length at most 3. Then G is Z3-connected.'Let G be a simple graph with|V(G)|=2n. G is called a k-edge deletable IM-extendable graph, if, for every F c E(G) with |F|=k, G-F is IM-extendable.In the first chapter, we mainly introduced the definition of three kinds graph-s including alternating group network ANn, alternating group graph AGn and 1-edge deletable connected IM-extendable graph. And some exist results, lemmas and theo-rems such as wheels, complete graphs, chordal graphs, triangularly connected graphs are gotten. In the second chapter the proofs of group connectivity for above mentioned graphs are given. For alternating group network graph AN4, by contracting subgraph which is isomorphic to k3, we can get the upper bound of its group connectivity is 4 and its lower bound is 2. Furthermore we prove its group connectivity is not equal to 3. For alternating group graph AG4, the proof method is similar to AN4. We prove the conclusion that:the group connectivity of both alternating group network graph AN4 and alternating group graph AG4 are 4.In the third chapter, the group connectivity of 4-regular, claw-free,1-edge deletable connected IM-extendable graphs are proved. We discussed its group connectivity by considering classification of its edges number among a fixed vertex's neighbor vertices. From the proof, we got the conclusion that:The group connectivity of 4-regular, claw-free,1-edge deletable connected IM-extendable graph is 3.The last chapter is the conclusion of this paper and open questions about alternating group graph AGn and alternating group network graph ANn.