Algebras of graph functions

Differential operators acting on functions defined on graphs by different studies do not form a consistent framework for the analysis of real or complex functions in the sense that they do not satisfy the Leibniz rule of any order. In this paper we propose a new family of operators that satisfy the Leibniz rule, and as special cases, produce the specific operators defined in the literature, such as the graph difference and the graph Laplacian. We propose a framework to define the order of a differential operator consistently using the Leibniz rule in Lie algebraic setting. Furthermore by identifying the space of functions defined on graph edges with the tensor product of node functions we construct a Lie bialgebra of graph functions and reinterpret the difference operator as a co-bracket. As an application, some explicit solutions of Schr\"odinger and Fokker-Planck equations are given.