Some Properties Of Bi-Cayley Graphs

The bipartite graph is playing a very important role in the study of graph theory. When people were studying bipartite graphs, they found a kind of graph with the following special property. That is for a bipartite graph X, Aut(X) exits a subgroup which acting regularly on the two parts of the graph X. We call this kind of graph X Bi-Cayley graph. In fact,that kind of graph can be constructed from group directly: let G is a finite group, S is a subset of G (possibly, contains i), the Bi-Cayley graph of G with respect to S is defined as the bipartite graph with vertex set G×{0,1} and edge set {(g, 0), (sg,1)∣g∈G, s∈S}, let X:=BCay(G,S).Bi-Cayley graph is standard double covering of (simple)Cayley graph, there are many similarities between them, and also many differences. For example, (simple)Cayley graph is vertex transitive, but Bi-Cayley graph may not vertex transitive (reference Lamma 2.2.10). All these special properties bring many difficulties for the study of Bi-Cayley graphs.We all know that the (simple)Cayley graph has been studied for a long history, the studies contain CI property, normal property and so on, and we have abundance results about it (reference [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and so on), but we have few results about Bi-Cayley graphs till now days, and representative results reference [15, 16], so it is still very important for the study of Bi-Cayley graphs.In this paper we mainly study the following two questions:1. The properties of Bi-Cayley graphs (such as the vertex transitive property and connect property) and the relations between Bi-Cayley graphs and (simple)Cayley graphs. And the last property may be useful for the study of Bi-Cayley graphs from the properties of (simple)Cayley graphs.2. The isomorphim problems of Bi-Cayley graphs, that is the BCI property of Bi- Cayley graphs which just like the CI property of (simple) Cayley graphs. We mainly study the BCI property of cyclic group of order p, and group of order p~i, pq(p, q are different prime numbers).