Abstract
In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph G = (V, E, d) is an undirected graph G = (V, E) associated with a distance function d: E -> [infinity, 8). By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between u and v is exactly d(u, v) for any edge uv. Now consider a random walk whose transitive probability from an vertex u to its neighbor v (a jump move along the edge uv) is proportional to w(uv) := F(d(u, v))/d(u, v) for some given function F(center dot). We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of kappa(x, y) using a so called *-coupling functions. The total curvature K(G) is defined to be the sum of Ricci curvatures over all edges of G. We proved the following theorems: if F(center dot) is a decreasing function, then K(G) >= 2 vertical bar V vertical bar - 2 vertical bar E vertical bar; if F(center dot) is an increasing function, then K(G) <= 2 vertical bar V vertical bar - 2 vertical bar E vertical bar. Both equations hold if and only if d is a constant function plus the girth is at least 6.
In particular, these imply a Gauss-Bonnet theorem for (un-weighted) graphs with girth at least 6, where the graph Ricci curvature is defined geometrically in terms of optimal transport.