On the spectral density function of the Laplacian of a graph

Let XX be a finite graph. Let |V||V| be the number of its vertices and dd be its degree. Denote by F1(X)F1(X) its first spectral density function which counts the number of eigenvalues ≤λ2≤λ2 of the associated Laplace operator. We provide an elementary proof for the estimate F1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λF1(X)(λ)−F1(X)(0)≤2⋅(|V|−1)⋅d⋅λ for 0≤λ<10≤λ<1 which has already been proved by Friedman (1996) [3] before. We explain how this gives evidence for conjectures about approximating Fuglede–Kadison determinants and L2L2-torsion.