The discrete Laplacian of a 2-simplicial complex

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.