Laplacian eigenvalues of equivalent cographs

Abstract

Let G and H be equivalent cographs with reductions and and suppose the vertices of and are labelled by the twin numbers of the k twin classes they represent. In this paper, we prove that G and H have at least Laplacian eigenvalues in common, where is the indices of the twin classes whose types are identical in G and H. This confirms the conjecture proposed by Abrishami [A combinatorial analysis of the eigenvalues of the Laplacian matrices of cographs [Master's thesis]. Johns Hopkins University; 2019. Available from: http://jscholarship.library.jhu.edu/bitstream/handle/1774.2/61684/ABRISHAMI-THESIS-2019.pdf]. We also show that no two nonisomorphic equivalent cographs are cospectral with relation to the Laplacian matrix.