  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.
