Convex functions on graphs: Sum of the eigenvalues

TL;DR

This paper explores bounds on sums of Laplacian eigenvalues in graphs, extending conjectures to weighted graphs and matrices, and relates convex functions on matrices to combinatorial properties.

Contribution

It generalizes conjectures about Laplacian eigenvalues to weighted graphs and symmetric matrices, and links convex functions on matrices to combinatorial bounds.

Findings

Extended conjectures to weighted graphs and matrices.

Established equivalence between bounds on convex functions of matrices.

Provided new bounds and relations for Laplacian eigenvalues.

Abstract

Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be the number of edges of $G$. A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $e(G)+\binom{k+1}{2}$ and $\sum_{i=1}^{k}d_{i}^{*}$, respectively, where $(d_{i}^{*})_{i}$ is the conjugate of the degree sequence $(d_i)_{i}$. We generalize these conjectures to weighted graphs and symmetric matrices. Moreover, among other results we show that under some assumptions, concave upper bounds on convex functions of symmetric real matrices are equivalent to concave upper bounds on convex functions of $(0,1)$ matrices.