  We investigate properties of spectrum of normalized Laplacian $\mathcal L$
for finite graphs over non-Archimedean ordered fields. We prove a Cheeger's
inequality for first non-zero eigenvalue. Then we describe properties of the
operator $\mathcal P=I-\mathcal L$, which is a generalization of transition
operator. We show that Cheeger estimate $\alpha_1\preceq \sqrt{1-h^2}$ for the
second largest eigenvalue of $\mathcal P$ is crucial for investigation of the
convergence of analogue of random walk to equilibrium over a non-Archimedean
ordered fields. We consider examples over the Levi-Civita field.
