29 research outputs found
The Graph Curvature Calculator and the curvatures of cubic graphs
We classify all cubic graphs with either non-negative Ollivier-Ricci
curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both
curvature notions that the non-negatively curved graphs are the prism graphs
and the M\"obius ladders. We also highlight an online tool for calculating the
curvature of graphs under several variants of these curvature notions that we
use in the classification. As a consequence of the classification result we
show, that non-negatively curved cubic expanders do not exist
Ricci-flat cubic graphs with girth five
We classify all connected, simple, 3-regular graphs with girth at least 5
that are Ricci-flat. We use the definition of Ricci curvature on graphs given
in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct.
Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all
edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are
the Petersen graph, the Triplex and the dodecahedral graph. This will correct
the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the
Triplex
Bakry-\'Emery and Ollivier Ricci Curvature of Cayley Graphs
In this article we study two discrete curvature notions, Bakry-\'Emery
curvature and Ollivier Ricci curvature, on Cayley graphs. We introduce Right
Angled Artin-Coxeter Hybrids (RAACHs) generalizing Right Angled Artin and
Coxeter groups (RAAGs and RACGs) and derive the curvatures of Cayley graphs of
certain RAACHs. Moreover, we show for general finitely presented groups that addition of relators does not lead to a
decrease the weighted curvatures of their Cayley graphs with adapted weighting
schemes