设$ G $是一个$ n $阶$ k $圈图, $ k $圈图为边数等于顶点数加$ k-1 $的简单连通图。$ \mu_{1}(G) $、$ \mu_{2}(G) $分别记为图$ G $的Laplace矩阵的最大特征值和次大特征值, 图$ G $的Laplace分离度定义为$ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $。本文研究了给定阶数的$ k $圈图的最大Laplace分离度, 并刻画了相应的极图, 其结果推广了已有当$ k=1, 2, 3 $时的结论。

Let $ G $ be an $ n $-order $ k $-cyclic graph. The $ k $-cyclic graph is a simply connected graph which the number of edges is equal to the number of vertices adding $ k-1 $. Let $ \mu_{1}(G) $ and $ \mu_{2}(G) $ be the largest eigenvalue and the second largest eigenvalue of the Laplacian matrix of $ G $, respectively. The Laplacian separator of graph $ G $ is defined as $ S_{L}(G)=\mu_{1}(G)-\mu_{2}(G) $. In this paper, we study the maximun Laplacian separator of $ k $-cyclic graph with given order, and characterize the according extremal graph. The result generalizes the existing conclusions when $ k=1, 2, 3 $.