Graphs Determined By Their Cycle Length Distributions

Complex networks are studied in many fields. And those complex networks are composed of some simple motifs. Cycles are the most important motifs in all sorts of networks. Some cycle subgraphs play an important role in many complex natural networks. So it is meaningful to study cycles in the general graphs. The cycle length distribution of graph G of ordern is(c₁,c₂,...,c_n), where c_i 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 distributions of graphs and obtain the following results(1)Let G=K_n-A,A(?)E(K_n),|A|=j,j≥6, n>1/2(j+ 1) +2 and K_n[A] has at most four paths of length 2, then G is determined by its cycle length distribution. (2) Let G = K_n,n-A, A(?) E[K_m=n,n),|A| = j,j≥6,n>1/2 j(j + l) + 2,K_n,n[A] has at most three paths of length 2, then G is determined by itscycle length distribution.