  For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can
act freely 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.
