On sums of graph eigenvalues

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.