# Condensed Ricci Curvature of Complete and Strongly Regular Graphs

## TL;DR

This paper investigates a modified Ricci curvature for graphs, providing rigidity results for complete graphs and explicit formulas for strongly regular graphs, revealing structural insights and limitations of curvature formulas based on graph parameters.

## Contribution

It introduces a rigidity theorem characterizing complete graphs via Ricci curvature and derives explicit formulas for strongly regular graphs, advancing understanding of graph curvature.

## Key findings

- Complete graphs have Ricci curvature > 1 if and only if they are complete.
- Explicit Ricci curvature formulas are derived for strongly regular graphs with girth 4 and 5.
- No simple formula exists for strongly regular graphs of girth 3 based solely on graph parameters.

## Abstract

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girth 4 and 5 using elementary methods. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth $3$ that is purely in terms of graph parameters.