The discrete Laplacian of a 2-simplicial complex

TL;DR

This paper introduces a Laplace operator on 2-simplicial complexes derived from graphs, explores geometric conditions for its self-adjointness, and advances the understanding of discrete differential operators on complex structures.

Contribution

It defines the Laplace operator on 2-simplicial complexes and establishes conditions for its essential self-adjointness based on $hi$-completeness, a new geometric criterion.

Findings

Defined the Laplace operator on 2-simplicial complexes

Established $hi$-completeness as a criterion for self-adjointness

Proved essential self-adjointness under geometric hypotheses

Abstract

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 $\chi$-completeness for the graphs, based on the cutoff functions. Moreover, we study essential self-adjointness of the discrete Laplacian from the $\chi$-completeness geometric hypothesis.