  In this paper, we explore the relationship between one of the most elementary
and important properties of graphs, the presence and relative frequency of
triangles, and a combinatorial notion of Ricci curvature. We employ a
definition of generalized Ricci curvature proposed by Ollivier in a general
framework of Markov processes and metric spaces and applied in graph theory by
Lin-Yau. In analogy with curvature notions in Riemannian geometry, we interpret
this Ricci curvature as a control on the amount of overlap between
neighborhoods of two neighboring vertices. It is therefore naturally related to
the presence of triangles containing those vertices, or more precisely, the
local clustering coefficient, that is, the relative proportion of connected
neighbors among all the neighbors of a vertex. This suggests to derive lower
Ricci curvature bounds on graphs in terms of such local clustering
coefficients. We also study curvature dimension inequalities on graphs,
building upon previous work of several authors.
