First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs

Abstract

Abstract. Following Escobar [Esc97] and Jammes [Jam15], we introduce two types of isoperimetric constants and give lower bound estimates for the first nontrivial eigenvalues of Dirichlet-to-Neumann operators on finite graphs with boundary respectively.

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

“…As we will see in the following examples, this lower bound reflects other aspects of the geometrical meaning of σ 1 than the one given by Theorem 1.3 of [8]. This is due to the fact that the diameter of the boundary does not depend on the total number of vertices of the graph.…”

“…This problem has b solutions σ 0 ≤ σ 1 ≤ ... ≤ σ b−1 , called the eigenvalues, and there exist b associated eigenfunctions v 0 , v 1 , ..., v b−1 which are mutually orthogonal for the bilinear form ·, · B and can be choosen such that v i B = 1. This affirmation results from the fact that Λ is a nonegative self-adjoint operator as explained in [8]. A proof using basic analysis and linear algebra tools is also given in [10].…”

“…By computation, we obtain that for this family of graphs, the lower bound of Theorem 1.3 in [8] tends to 0 as n goes to infinity. So, in this case where the diameter is unbounded but the diameter of the boundary is bounded, the Jammes-type Cheeger estimate fails to see that σ 1 is bounded.…”

“…The eigenvalues of this problem are the same as the eigenvalues of the discrete Dirichlet-to-Neumann operator defined by [8] in the case that the weights of the edges are all 1 and the degree of the boundary vertices is 1.…”

Evolution problems of Leray–Lions type with nonhomogeneous Neumann boundary conditions in metric random walk spaces

“…The approaches in [11] and [12] have been unified in [2]. Conditions like (1.4) were also introduced for graphs by Chung and Yau in [9] and [10] (see also [17] and [18]) for the study of the eigenvalues of the graph Laplacian operator. Namely, let G = (V (G), E(G)) be a finite weighted discrete connected graph (see Example 1.2) and Ω ⊂ V (G) be a set of vertices of G, their work comprises the study the eigenvalues of the graph Laplacian operator given by 1 d x y∈V (u(y) − u(x))w xy , x ∈ Ω, under the following Neumann boundary condition…”