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arXiv · 2404.15594

Curvature, diameter and signs of graphs

Wei Chen·Shiping Liu
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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.

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Wei Chen, Shiping Liu. 2024-04-24. Curvature, diameter and signs of graphs. https://arxiv.org/abs/2404.15594

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