On sums of graph eigenvalues

TL;DR

This paper develops variational techniques to establish upper bounds for sums of eigenvalues of matrices associated with finite graphs, generalizing known inequalities and providing conditions related to graph embeddability in lattices.

Contribution

It introduces new upper bounds for sums of eigenvalues of graph matrices, extending Fiedler's inequality and linking spectral sums to graph embeddability in lattices.

Findings

Derived upper bounds for sums of eigenvalues of graph matrices.

Generalized Fiedler's inequality to sums of the smallest or largest k eigenvalues.

Provided necessary conditions for graph embeddability in  lattices based on spectral sums.

Abstract

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices. Among our results, we generalize an inequality of Fiedler for the extreme eigenvalues of the graph Laplacian to a bound on the sums of the smallest (or largest) k such eigenvalues, k < n. Furthermore, if lambda_j are the eigenvalues of the positive graph Laplacian H, in increasing order, on a finite graph with |V| vertices and |E| edges which is isomorphic to a subgraph of the \nu-dimensional infinite cubic lattice, then the spectral sums obey a Weyl-type upper bound, a simplification of which reads $$ \sum_{j=1}^{k-1}{\lambda_j} \le \frac{\pi^2 |\mathcal{E}|}{3} \left(\frac{k}{|\mathcal{V}|} \right)^{1+\frac{2}{\nu}} $$ for each k < |V|. This and related estimates for sums of lambda_j^2 provide a family of necessary conditions for the embeddability of the graph in a lattice of dimension \nu or less.