Eigenvalue bounds of the Kirchhoff Laplacian

TL;DR

This paper establishes new bounds on the eigenvalues of the Kirchhoff Laplacian for graphs and quivers, linking them to vertex degrees, and provides improved inequalities for these eigenvalues.

Contribution

It introduces novel upper bounds for eigenvalues of the Kirchhoff Laplacian based on vertex degrees and extends lower bound inequalities to quivers.

Findings

Eigenvalues are bounded above by sums of vertex degrees.

The bounds apply to graphs and quivers.

Improved lower bounds are provided for quivers.

Abstract

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.