On the spectral density function of the Laplacian of a graph

TL;DR

This paper establishes an upper bound on the spectral density function of a graph's Laplacian, providing insights into spectral approximation conjectures related to Fuglede-Kadison determinants and L^2-torsion.

Contribution

It proves a new estimate for the spectral density function of graph Laplacians, linking spectral properties to graph parameters and supporting conjectures in spectral graph theory.

Findings

Proves an inequality: F_1(X)(lambda) - F_1(X)(0) ≤ 2·E·d·lambda for 0 ≤ lambda < 1

Provides evidence for conjectures on approximating Fuglede-Kadison determinants

Connects spectral density estimates with algebraic invariants of graphs

Abstract

Let X be a finite graph. Let E be the number of its edges and d be its degree. Denote by F_1(X) its first spectral density function which counts the number of eigenvalues less or equal to lambda^2 of the associated Laplace operator. We prove the estimate F_1(X)(lambda) - F_1(X)(0) le 2 cdot E cdot d cdot lambda for 0 le lambda < 1. We explain how this gives evidence for conjectures about approximating Fuglede-Kadison determinants and L^2-torsion.