On the Laplacian Integral of \((k-1)\)-Cyclic Graphs

Abstract

A graph is termed Laplacian integral if its Laplacian spectrum comprises integers. Let \(\theta(n_1, n_2, \ldots, n_k)\) be a generalized \(\theta\)-graph (see Figure 1). Denote by \(\mathcal{G}_{k-1}\) the set of \((k-1)\)-cyclic graphs, each containing some generalized \(\theta\)-graph \(\theta(n_1, n_2, \ldots, n_{k})\) as its induced subgraph. In this paper, we establish an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1), from which we identify all Laplacian integral graphs in the class \(\mathcal{G}_{ k-1}\) (Theorem 3.2).