  We prove the following estimate for the spectrum of the normalized Laplace
operator $\Delta$ 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 $k\leq \lambda_1\leq \ldots \leq
\lambda_{N-1}\leq 2-k$. For sufficiently large $t$ we show that, 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 $k\leq 0$. 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.
