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arXiv · 2506.10492

Repelling curvature via $ε-$repelling Laplacian on positive connected signed graphs

Yong Lin·Shi Wan
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Abstract

The paper defines a positive semidefinite operator called $ε-$repelling Laplacian on a positive connected signed graph where $ε$ is an arbitrary positive number less than a constant $ε_0$ related to the graph's consensus problem. Then we investigate the upper bound of the second smallest eigenvalue of $ε-$repelling Laplacian. Besides, we use the pseudoinverse of $ε-$repelling Laplacian to construct a simplex as well as $ε-$repelling cost whose square root turns out to be a distance among the vertices of the simplex. We also extend the node and edge resistance curvature proposed by K.Devriendt et al. to node and edge $ε-$repelling curvature and derive the corresponding Lichnerowicz inequalities on any positive connected signed graph. Moreover, it turns out that edge $ε-$repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is $ε-$repelling cost rather than the length of the shortest path.

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Yong Lin, Shi Wan. 2025-06-12. Repelling curvature via $ε-$repelling Laplacian on positive connected signed graphs. https://doi.org/10.1007/s12220-026-02420-1

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