Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence

Abstract

In this paper we consider particular graphs defined by $\overline{\overline{\overline{K_{{\alpha }_1}}\cup K_{{\alpha }_2}}\cup \cdots \cup K_{{\alpha }_k}}$, where $k$ is even, $K_{\alpha }$ is a complete graph on $\alpha$ vertices, $\cup$ stands for the disjoint union and an overline denotes the complementary graph. These graphs do not contain the $4$-vertex path as an induced subgraph, i.e., they belong to the class of cographs. In addition, they are iteratively constructed from the generating sequence $({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)$. Our primary question is what invariants or graph properties can be deduced form a given sequence. In this context, we compute the Lapacian eigenvalues and the corresponding eigenspaces, and derive a lower and an upper bound for the number of distinct Laplacian eigenvalues. We also determine the graphs under consideration with a fixed number of vertices that either minimize or maximize the algebraic connectivity (that is the second smallest Laplacian eigenvalue). The clique number is computed in terms of a generating sequence and a relationship between it and the algebraic connectivity is established.