Monotonicity of Steklov eigenvalues on graphs and applications

Abstract

In this paper, we obtain monotonicity of Steklov eigenvalues on graphs which as a special case on trees extends the results of He-Hua [arXiv: 2103.07696] to higher Steklov eigenvalues and gives affirmative answers to two problems proposed in He-Hua [arXiv: 2103.07696]. As applications of the monotonicity of Steklov eigenvalues, we obtain some estimates for Steklov eigenvalues on trees generalizing the isodiametric estimate for the first positive Steklov eigenvalues on trees in He-Hua [arXiv:2011.11014].

Minimal Steklov eigenvalues on combinatorial graphs

“…To extend Friedman's theory of nodal domains for Laplacian eigenfunction to Steklov eigenfunctions, we need to consider Steklov eigenvalues with vanishing Dirichlet boundary data as in [15,27]. So, we introduce the notion of graph with boundary and Dirichlet boundary.…”

“…Heuristically, the wedge-sum of two brooms on their roots is called a dumbbell. For the definition of wedge-sum of graphs, see [14,27]. Definition 2.5.…”

“…For example, in [13,16,17,12,14,20,26], the authors studied isoperimetric control of Steklov eigenvalues. In [15,27], the authors studied monotonicity of Steklov eigenvalues. In [19], the author gave some interesting upper or lower bounds.…”

“…In this paper, motivated by the works of Fraser and Schoen in the smooth setting and Friedman's works [10,11] for Laplacian eigenvalues on combinatorial graphs, and also motivated by the works [14,15] of He and Hua, and our previous work [27], we consider extremal problems on combinatorial graphs. Let's first recall some preliminary notions for graphs.…”

“…Our strategy to solve the extremal problem for Steklov eigenvalues on combinatorial finite graphs is similar to the strategy by Friedman [11] solving the corresponding extremal problems for Laplacian eigenvalues. First by the following simple monotonicity of Steklov eigenvalues that is different with the monotonicity in [15,27], we reduce the extremal problems to trees. Theorem 1.2.…”