Minimal Steklov eigenvalues on combinatorial graphs

TL;DR

This paper investigates extremal Steklov eigenvalues on combinatorial graphs, extending Friedman's theory to identify minimal eigenvalues achieved by specific graph structures.

Contribution

It extends Friedman's nodal domain theory to Steklov eigenfunctions and characterizes graphs minimizing the Steklov eigenvalues.

Findings

Minimum Steklov eigenvalues are attained by star graphs with minimal broom arms.

When divisibility conditions are met, the extremal graph is a regular comb with minimal broom teeth.

The results provide a combinatorial analogue to classical Laplacian eigenvalue extremal problems.

Abstract

In this paper, we study extremal problems of Steklov eigenvalues on combinatorial graphs by extending Friedman's theory [Duke Math. J. 69 (1993), no. 3, 487--525] of nodal domains for Laplacian eigenfunctions to Steklov eigenfunctions, and solve an extremal problem for Steklov eigenvalues on combinatorial graphs that is an analogue of the extremal problem solved by Friedman [Duke Math. J. 83 (1996), no. 1, 1--18.] for Laplacian eigenvalues. More precisely, we mainly show that the minimum of the $i^{\rm th}$ Steklov eigenvalue on a connected combinatorial graph with $n$ vertices is essentially attained by a star with each arm a minimal broom when $i\not|n$, and attained by a regular comb with each tooth a minimal broom when $i|n$.