On the Laplacian spectrum of k-symmetric graphs

Abstract

For some positive integer k, if the finite cyclic group Zk can freely act on a graph G, then we say that G is k-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected k-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of k-symmetric graphs in which all Laplacian eigenvalues are integers.