Bipartite Graphs Determined By Their Cycle Length Distributions

Complex networks are studied aross many fields. And those complex networks are composed of some simple motifs. Cycle is the most important mtoifs in all sorts of networks. Some cycle subgraphs perform important functions in many of the complex networks that occure in the nature. Therefore, there's some important meaning in studying cycles in the general graphs. The cycle length distribution of graph G of order n is (c1,c2,..., cn), where c, is the number of cycles of length i. If a graph is determined by its cycle length distribution, it'll have more special meaning. In this paper we study the cycle length distribution of bipartite graphs and obtain the following result: (1) If |A|≤1, and n≤r≤min{n+6,2n-3}, then Kn,r-A is determined by its cycle length distribution . (2) If |A|=2, and n≤r≤min{n+6,2n-5}, then Kn,r-Ais determined by its cycle length distribution . (3) If |A|=3, and n≤r≤min{n+6,2n-7},then Kn,r-A is also determined by its cycle length distribution. Here A E(Kn,r).