论文标题
通过平衡度量在图上的曲率
Curvature on Graphs via Equilibrium Measures
论文作者
论文摘要
我们在有限的组合图上引入了曲率的概念。可以通过求解方程式系统来轻松计算它。我们表明，曲率下面的图形以$ k> 0 $为$ \ mbox {diam}（g）（g）\ leq 2/k $（一个帽子的理论），$ \ mbox {diam}（diam}（g）（g）= 2/k $表示$ g $ g $ a $ g $ g $ a $ ge a $ a ceptator geap a a a is a ceeng a a a ceppator a a an ceppation a an ceppeptral a a and ceppeptian nisepperta k/（2n）$（Lichnerowicz定理）。它是针对几个图的家族计算的，通常与曲面或lin-lu-yau曲率相吻合。 von Neumann minimax定理在证明中以突出的特征。
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 $K>0$ have diameter bounded by $\mbox{diam}(G) \leq 2/K$ (a Bonnet-Myers theorem), that $\mbox{diam}(G) = 2/K$ implies that $G$ has constant curvature (a Cheng theorem) and that there is a spectral gap $λ_1 \geq K/(2n)$ (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.
