Curvature on graphs via equilibrium measures

• Published in: Journal of graph theory Vol. 103; no. 3; pp. 415 - 436
• Main Author: Steinerberger, Stefan
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.07.2023
• Subjects: Bonnet–Myers; Combinatorial analysis; Computation; Curvature; Diameters; graph; Graphs; Lichnerowicz; Minimax technique; minimax theorem; Theorems
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22925

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by K>0 $K\gt 0$ have diameter bounded by diam(G)≤2∕K $\text{diam}(G)\le 2\unicode{x02215}K$ (a Bonnet–Myers theorem), that diam(G)=2∕K $\text{diam}(G)=2\unicode{x02215}K$ implies that G $G$ has constant curvature (a Cheng theorem) and that there is a spectral gap λ1≥K∕(2n) ${\lambda }_{1}\ge K\unicode{x02215}(2n)$ (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.