图的圈长分布和圈长分布唯一的图

The Cycle Length Distribution of a Graph and Unique Cycle Length Distribution Graphs

阶为ｎ的图Ｇ的圈长分布是指序列（ｃ１，ｃ２，…，ｃｎ），其中ｃｉ是Ｇ中长为ｉ的圈数．若不存在，使Ｇ’与Ｇ有相同的圈长分布，则称图Ｇ是圈长分布唯一图．本文确定了Ｋｎ－Ａ（｜Ａ｜＝ｊ，ｎ≥｜Ａ｜＋３）的最小、最大的４圈和５圈数．证明了当ｎ≥９时，Ｋｎ－Ａ（｜Ａ｜＝４）以及当ｎ≥１４时，Ｋｎ－Ａ（｜Ａ｜＝５）都是圈长分布唯一图．

The cycle length distribution of order n is (c1,c2, …,cn ),where ci is the number of cycles of length i. If there does nut exist G' (G' G, whose cycle length distribution is the same as that of G, then G is called a unique cycle length distribution graph.In this paper, the maximum and minimum numbers of cycles of length 4 and 5 on Kn-A (|A| = j,n ≥|A|+ 3) are obtained. It is also proven that Kn-A (|A|=4, n ≥9) and Kn.- A (|A|= 5, n ≥14) are unique cycle length distribution graphs.