The Expand And Application Of Erd(?)s-Rényi Model In Cayley Graphs

The paper researches two problems as follows with the probability method applied in Cayley graphs by professor Meng.(一) Let G be a finite group of order n and S (possibly, contains the identity element) be a subset of G. The Bi-Cayley graph BC(G, S) of G is a bipartite graph with vertex set G×{0,1} and edge set {{(g, 0), (gs,1)}|g∈G,s∈S}. Let p (0 < p < 1) be a fixed number. We define B = {X = BC{G,S),S (?) G) as a sample space and assign a probability measure by requiring P_?)X) =p^kq^n-k. for X = BC(G,S) with |S| = k. Here it is shown that the probability of the set of Bi-Cayley graph of G with diameter 3 approaches 1 as the order n of G approaches infinity.(二) Let G be a finite group of order n and S be a subset of G not containing the identity element. Let p (0 < p < 1) be a function of n. We define the set of all labelled Cayley graphs C(G,S),S (?) G \ {1}, S^-1 = S and Cayley digraphs D(G, S) S (?) G \ {1} (0 <| S|≤|G| -1) as a sample space and assign a probability measure by requiring P_r(a∈S) = p for any a∈G \ {1}. Here it is shown that if p; << n^?,ω(C(G.S)) < 4 almost surely; if p >> n^?,ω(C{G,S))≥4 almost surely. If p <?,ω^*(D(G.S)) < 4 almost surely; if p >> n^?,ω^*(D(G,S))≥4 almost surely.