Ricci curvature, circulants, and a matching condition

The Ricci curvature of graphs, as recently introduced by Lin, Lu, and Yau following a general concept due to Ollivier, provides a new and promising isomorphism invariant. This paper presents a simplified exposition of the concept, including the so-called logistic diagram as a computational or visualization aid. Two new infinite classes of graphs with positive Ricci curvature are identified. A local graph-theoretical condition, known as the matching condition, provides a general formula for Ricci curvatures. The paper initiates a longer-term program of classifying the Ricci curvatures of circulant graphs. Aspects of this program may prove useful in tackling the problem of showing when twisted tori are not isomorphic to circulants.