On the Laplacian integral (k - 1)-cyclic graphs

A graph is called Laplacian integral if its Laplacian spectrum consists of integers. Let theta(n(1), n(2) ..., n(k)) be a generalized theta-graph (see Figure 1). Denote by g(k-1) the set of (k - 1)-cyclic graphs each of them contains some generalized theta-graph theta(n(1), n(2), ..., n(k)) as its induced subgraph. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1), from which we identify all the Laplacian integral graphs in the class g(k-1) (Theorem 3.2).