Ricci Curvature Formula: Applications to Bonnet-Myers Sharp Irregular Graphs

In this paper, we establish a simple formula for computing the Lin-Lu-Yau Ricci curvature on graphs. For any edge $xy$ in a simple locally finite graph $G$, the curvature $\kappa (x,y)$ can be expressed as a cost function of an optimal bijection between two blow-up sets of the neighbors of $x$ and $y$. Utilizing this approach, we derive several results including a structural theorem for the Bonnet-Myers sharp irregular graphs of diameter $3$ and a theorem on $C_3$-free Bonnet-Myers sharp graphs.