A brief note about p-curvature on graphs

TL;DR

This paper explores the properties of p-curvature on graphs under Wang's $CD_p(m,K)$ condition, revealing how it varies with p and highlighting differences from classical curvature concepts, especially in Cartesian products.

Contribution

It extends the study of curvature dimension conditions to the p-Laplacian on graphs, providing explicit examples and analyzing how p affects curvature behavior and product properties.

Findings

p-curvature is non-negative at some vertices for p ≥ 2

p-curvature approaches -∞ for 1 < p < 2

non-negative curvature preservation under Cartesian products fails for p > 2

Abstract

In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $\Delta_p$ for $p>1$ and is an extension of the classical Bakry-\'Emery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1<p<2$. In addition, we observe that a crucial property of $\Gamma_2$ on Cartesian products does no longer hold for $\Gamma_2^p$ in the case of $p > 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$.