Lower bounds for the first eigenvalue of the Steklov problem on graphs

TL;DR

This paper establishes new lower bounds for the first non-zero Steklov eigenvalue on connected graphs, emphasizing boundary properties over overall graph diameter, with sharpness results and comparisons to Cheeger inequalities.

Contribution

It introduces boundary-dependent lower bounds for Steklov eigenvalues on graphs, including weighted cases and asymptotic sharpness analysis.

Findings

Lower bounds depend on boundary extrinsic diameter, not graph diameter.

Bounds are sharp for boundary size of 2.

Comparison with Cheeger inequality shows advantages of the new bounds.

Abstract

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weigthed graphs and compare our result to the Cheeger inequality.