Higher Steklov eigenvalues of graphs on surfaces

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows. Our result can be regarded as a discrete analogue of Karpukhin’s bound in spectral geometry. Moreover, this result implies the upper bound of higher Laplacian eigenvalues given by Kelner, Lee, Price and Teng (Geom. Funct. Anal., 2011).