Uniqueness of Cycle Length Distributions of Certain Bipartite Graphs $K_{n,n+7}-A(|A|\leq3)$

Abstract:

      The cycle length distribution of a graph of order $n$ is denoted by $(c_1,c_2,\ldots,c_n)$, where $c_i$ is the number of cycles of length $i$. In this paper, we obtain that a graph $G$ is uniquely determined by its cycle distribution if: (1) $G=K_{n,n+7}$ $(n\geq10)$; or (2) $G=K_{n,n+7}-A~(|A|=1,n\geq12)$; or (3) $G=K_{n,n+7}-A~(|A|=2,n\geq14)$; or (4) $G=K_{n,n+7}-A~(|A|=3,n\geq16)$, where $A\subseteq E(K_{n,n+7})$.