Professor Xiaonan Liu Discrete Math Seminar Talk

Title: Curvature on Graphs: From Local Geometry to Global Structure

Abstract: Inspired by Ricci curvature in geometry, Lin–Lu–Yau curvature approaches these questions in graphs through random walks and optimal transport, providing a connection between geometry, probability, and combinatorics. In this talk, I will explain the motivation behind Lin–Lu–Yau curvature, introduce its definition through concrete examples, and explore several applications. We will see how a positive lower bound on curvature controls a graph’s diameter and Laplacian eigenvalues, yielding discrete analogues of classical geometric theorems. I will also discuss examples from recent joint work showing how positive curvature, together with combinatorial restrictions, can sharply constrain the maximum degree and the number of vertices.