A brief note about p-curvature on graphs

In this paper, we consider Wang’s $C{D}_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian ${\mathrm{\Delta }}_p$ for $p>1$ and is an extension of the classical Bakry-Émery $CD(m,K)$ 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 $p\ge 2$, while it approaches to $-\mathrm{\infty }$ in the case of $1<p<2$. In addition, we observe that a crucial property of ${\mathrm{\Gamma }}_2$ on Cartesian products does no longer hold for ${\mathrm{\Gamma }}_2^p$ 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 $p>2$.