Abstract：Circulant graphs are an important class of network topology. Let G be a simple graph with n vertices, let A be the adjacency matrix of G, and λ1,λ2,…,λn be the eigenvalues of graph G. As a kind of centrality of complex networks, the resolvent Estrada index of G is defined as EEr(G)=((1-λi)/(n-1))-1. By Ramanujan's sum, using the Euler function and Mobius function, we characterize the lower bound of resolvent Estrada index of circulant graph, and obtain some computational formulas of integral circulant graphs.
[1] CHEN Xiaodan, QIAN Jianguo. Bounding the resolvent Estrada index of a graph[J]. Journal of Mathematical Study,2012(2):159-166.
[2] CHEN Xiaodan, QIAN Jianguo. On resolvent Estrada index[J]. Match Commun Math Comput Chem,2015,73:163-174.
[3] IVAN G, BORIS F, CHEN X. Graphs with smallest resolvent Estrada indices[J]. Match Commun Math Comput Chem,2015,73:267-270.
[4] SO W. Integral circulant graphs[J]. Discrete Mathematics,2006,306:153-158.
[5] DAVIS P J. Circulant Matrices[M]. New York: John Wiley & Sons,1979.
[6] KLOTZ W, SANDER T. Some properties of unitary Cayley graphs[J]. The Electronic Journal Combinatorics,2007,14(1):697-714.
2016, Vol. 43 
（（1-λi）/（n-1））-1，这里，λ1，λ2，…，λn是图G的邻接矩阵的特征值.利用Ramanujan和，借助于欧拉函数和莫比乌斯函数，讨论并得到了循环图的预解Estrada指标的下界以及整循环图的预解Estrada指标的几个计算公式.
((1-λi)/(n-1))-1. By Ramanujan's sum, using the Euler function and Mobius function, we characterize the lower bound of resolvent Estrada index of circulant graph, and obtain some computational formulas of integral circulant graphs.