Factorization and recursion relations of the matching and characteristic polynomials of periodic polymer networks

Abstract

Recent developments in the analysis of mathematical structure of the matching and characteristic polynomials of linear and cyclic periodic polymer networks are surveyed, especially on the newly found efficient algorithms and techniques for deriving their recursion relations and factorization expressions. Advantages and disadvantages of these two polynomials for manipulating large networks are compared and discussed with examples. Contrary to the case of singly connected polymer networks, only a few useful mathematical properties are shown to be found for doubly connected polymer networks. Linear and cyclic fence graphs are proposed to be defined instead of the conventional definitions of the so-called Huckel and Mobius ladder graphs, so that simpler and more useful mathematical relations hold for their matching polynomials.