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

TL;DR

This paper introduces the $$-repelling Laplacian for positive signed graphs, analyzing its eigenvalues, constructing a related simplex, and extending curvature concepts with new inequalities.

Contribution

It defines the $$-repelling Laplacian, explores its spectral properties, and extends curvature notions to signed graphs with new inequalities.

Findings

Upper bound of the second smallest eigenvalue of $$-repelling Laplacian

Construction of a simplex using the pseudoinverse of $$-repelling Laplacian

Extension of curvature concepts with derived Lichnerowicz inequalities

Abstract

The paper defines a positive semidefinite operator called $\epsilon-$repelling Laplacian on a positive connected signed graph where $\epsilon$ is an arbitrary positive number less than a constant $\epsilon_0$ related to the graph's consensus problem. Then we investigate the upper bound of the second smallest eigenvalue of $\epsilon-$repelling Laplacian. Besides, we use the pseudoinverse of $\epsilon-$repelling Laplacian to construct a simplex as well as $\epsilon-$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 $\epsilon-$repelling curvature and derive the corresponding Lichnerowicz inequalities on any positive connected signed graph. Moreover, it turns out that edge $\epsilon-$repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is $\epsilon-$repelling cost rather than the length of the shortest path.