循环图的Kirchhoff指标

The Kirchhoff index of circulant graphs

图G的Kirchhoff指标定义为G中所有点对之间的电阻距离之和,记为Kf(G).图G为循环图,如果图G的邻接矩阵是循环矩阵;图G为整谱图,若它的特征值全为整数.该文利用循环图的Laplacian谱,讨论了循环图的Kirchhoff指标下界;借助Ramanujan和,利用Euler函数和Mobius函数,得到了一个关于整循环图的Kirchhoff指标的简便计算公式.这样无须求出整循环图的特征值,也可求整循环图的Kirchhoff指标.

The Kirchhoff index,Kf（G）,is the sum of resistance distances between all pairs of vertices in G.A graph G is called circulant graph if it has a circulant adjacency matrix.A graph G is called an integral graph if it has an integral spectrum.In this paper,using Laplacian spectrum of circulant graphs,we discuss and give a new lower bound for the Kirchhoff index.According to Ramanujan sums,applying the Euler function and the Mobius function,we obtain a simple formula for the Kirchhoff index of integral circulant graphs.Then we can calculate the Kirchhoff index,and we do not calculate its eigenvalues.