An interlacing property of the signless Laplacian of threshold graphs

TL;DR

This paper demonstrates an interlacing property of the signless Laplacian eigenvalues with vertex degrees in threshold graphs and verifies the signless Brouwer conjecture for this class.

Contribution

It establishes a new interlacing property for the signless Laplacian eigenvalues of threshold graphs and confirms the conjecture for this specific graph class.

Findings

Eigenvalues of the signless Laplacian interlace with vertex degrees in threshold graphs.

The signless Brouwer conjecture holds for threshold graphs.

Provides a bound on the sum of the largest eigenvalues in threshold graphs.

Abstract

We show that for threshold graphs, the eigenvalues of the signless Laplacian matrix interlace with the degrees of the vertices. As an application, we show that the signless Brouwer conjecture holds for threshold graphs, i.e., for threshold graphs the sum of the k largest eigenvalues is bounded by the number of edges plus k + 1 choose 2.