Operations on graph functions and spectral properties of compositions of reflections

The article deals with operators acting on spaces of graph functions. Graph-theoretic methods are used to find the properties of the introduced operators. These properties show that the introduced operators are discrete analogues of differentiation and integration. The values of operators on some important graph functions are found. A method of using operators to study graph functions and methods of expressing some functions through others are developed. The characteristic polynomials of the Coxeter transformation is considered. Its coefficients can be expressed in terms of simple graph functions. With the help of the developed methodology a method of finding such expressions is proposed. The results of the article can be used to find spectral characteristics of compositions of reflections. These methods are simple and convenient to use.