In this paper, we introduce the notion of oriented faces especially triangles in a connected oriented locally finite graph. This framework then permits to define the Laplace operator on this structure of the 2-simplicial complex. We develop the notion of χ-completeness for the graphs, based on the cut-off functions. Moreover, we study essential self-adjointness of the discrete Laplacian from the χ-completeness geometric hypothesis . Résumé. Dans cet article, nous introduisons la notion de faces orientées et plus particulièment de triangles dans un graphe connexe orienté localement fini. Ce cadre permet alors de définir l'opérateur de Laplace sur cette structure d'un 2-complexe simplicial. Nous développons la notion de χ-complétude pour les graphes, basée sur les fonctions de coupure. De plus, nousétudions le caractère essentiellement auto-adjoint du Laplacien discretà partir de l'hypothèse géométrique χ-complétude.
In the context of transient graphs, we study the first Steklov eigenvalue σ0(Ω) of an infinite subgraph with finite boundary (Ω, B) of the integer lattice Zn. We focus in this paper on finding lower bounds using the technique of discretization of smooth compact Riemannian manifolds with cylindrical boundary. These bounds essentially depend on the discretization of the sphere Sn ⊂ Rn+1 with two identical boundaries’ isometrics to {1} × Sn−1 through quasi-isometries. As a consequence, if n ≥ 4 and the boundary B considered as a finite subset of Zd where 1 ≤ d ≤ n − 3, we show that σ0(Ω)|B| 1/n−1 tends to infinity as the cardinal of B tends to infinity. Moreover, if n ≥ 3 and the boundary B is a sphere, we prove that the first Steklov eigenvalue tends to zero proportionally to 1/|B| 1/n−1 as the radius tends to infinity and that σ0(Ω)|B| 1/n−1 is bounded.
2010 Mathematics Subject Classification. 39A12, 05C63, 47B25, 05C12, 05C50.
Our goal in this paper is to find an estimate for the spectral gap of the Laplacian on a two-simplicial complex consisting on a hypergraph of a complete graph. An upper estimate is given by generalizing the Cheeger constant. The lower estimate is obtained from the first non-zero eigenvalue of the discrete Laplacian acting on the functions of certain sub-graphs.
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