Laplace and Dirac Operators on Graphs

TL;DR

This paper introduces new variations of Laplace and Dirac operators on graphs, explores their applications to graph-theoretic Schrödinger and Dirac equations, and provides combinatorial interpretations and identities related to these operators.

Contribution

It presents novel graph-based versions of Laplace and Dirac operators, along with solutions and identities, expanding their theoretical framework and applications.

Findings

Provided combinatorial interpretations for solutions of graph Schrödinger and Dirac equations.

Proved gluing identities for Dirac operators on lattice graphs.

Developed graph Clifford algebra frameworks.

Abstract

Discrete versions of the Laplace and Dirac operators haven been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schr\"odinger and Dirac equation. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras.