Laplacian integrality in P4-sparse and P4-extendible graphs

TL;DR

This paper characterizes which P4-sparse and P4-extendible graphs have Laplacian matrices with all integer eigenvalues, extending known results from cographs.

Contribution

It provides a complete characterization of L-integral graphs within P4-sparse and P4-extendible classes, expanding understanding beyond cographs.

Findings

P4-sparse graphs are L-integral if and only if they are cographs.

P4-extendible graphs are L-integral precisely when they are cographs.

The characterization extends the class of graphs known to have integral Laplacian spectra.

Abstract

Let G be a simple graph and L = L(G) the Laplacian matrix of G. G is called L-integral if all its Laplacian eigenvalues are integer numbers. It is known that every cograph, a graph free of P4, is L-integral. The class of P4-sparse graphs and the class of P4-extendible graphs contain the cographs. It seems natural to investigate if the graphs in these classes are still L-integral. In this paper we characterized the L-integral graphs for both cases, P4-sparse graphs and P4-extendible graphs.