Laplacian eigenvalues and eigenspaces of cographs generated by finite sequence

Abstract

In this paper we consider particular graphs defined by (Formula Presented.), where k is even, Kα is a complete graph on α vertices, ∪ 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 (α1,α2,…,αk). Our primary question is which invariants or graph properties can be deduced from 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.