On summation over paths in spectral graph theory

Various types of summation over paths occur in graph theory, for example, in situations, where graphs arise in the form of Feynman diagrams in perturbation theory. Another example is the situation where a molecule of hydrocarbon is described by a graph modeling the structure of the molecule. This situation generated the spectral theory of graphs, where the summation of characteristic polynomials over simple paths is commonly used. Such sums are considered in this paper. We prove that such a sum is expressed in terms of the polynomials of four subgraphs and illustrate some applications of this result.