Curvature, diameter and signs of graphs

TL;DR

This paper establishes eigenvalue-diameter bounds for non-negatively curved signed graphs, leading to volume estimates and Li-Yau inequalities, using gradient estimates and nodal domain techniques, with extensions to nonlinear Laplacians.

Contribution

It introduces a Li-Yau type eigenvalue-diameter estimate for signed graphs and develops new methods involving nodal domain walks, extending results to nonlinear Laplacians.

Findings

Eigenvalues are bounded below by 1/D^2 for non-negatively curved signed graphs.

Volume estimates relate frustration index and diameter.

Two-sided Li-Yau estimates are obtained for triangle-free graphs.

Abstract

We prove a Li-Yau type eigenvalue-diameter estimate for signed graphs. That is, the nonzero eigenvalues of the Laplacian of a non-negatively curved signed graph are lower bounded by $1/D^2$ up to a constant, where $D$ stands for the diameter. This leads to several interesting applications, including a volume estimate for non-negatively curved signed graphs in terms of frustration index and diameter, and a two-sided Li-Yau estimate for triangle-free graphs. Our proof is built upon a combination of Chung-Lin-Yau type gradient estimate and a new trick involving strong nodal domain walks of signed graphs. We further discuss extensions of part of our results to nonlinear Laplacians on signed graphs.