Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator

We prove the following estimate for the spectrum of the normalized Laplace operator $\mathrm{\Delta }$ on a finite graph $G$, \begin{equation*}1- (1- k[t])^{\frac{1}{t}}\leq \lambda_1 \leq \cdots \leq \lambda_{N-1}\leq 1+ (1- k[t])^{\frac{1}{t}}, ,\forall ,,$\text{integers}$,, t\geq 1. \end{equation*} Here $k[t]$ is a lower bound for the Ollivier-Ricci curvature on the neighborhood graph $G[t]$, which was introduced by Bauer-Jost. In particular, when $t=1$ this is Ollivier’s estimates $k\leq \lambda_1\leq \ldots \leq \lambda_{N-1}\leq 2-k$\mathrm{.}Forsufficientlylarge$t$weshowthat,unless$G$ is bipartite, our estimates for ${\lambda }_1$ and ${\lambda }_{N-1}$ are always nontrivial and improve Ollivier’s estimates for all graphs with $k\le 0$. By definition neighborhood graphs are weighted graphs which may have loops. To understand the Ollivier-Ricci curvature on neighborhood graphs, we generalize a sharp estimate of the Ricci curvature given by Jost-Liu to weighted graphs with loops and relate it to the relative local frequency of triangles and loops.