Generalized Wheland polynomial for large graphs

Abstract

We report the generalized Wheland polynomial for acyclic graphs depicting polyenes havingn = 10 carbon atoms. We consider the problem of deriving generalized Wheland polynomials for larger chains by recursion. The recursion Wh(n + l;x) =, Wh(n; x) + (1 −x)Wh(n − 1;x) allows one to find the next larger generalized Wheland polynomial for a chain having an even number of vertices by knowing generalized Wheland polynomials of chains having fewer vertices. The recursion, however, does not allow one to predict the generalized Wheland polynomial for a chain having an odd number of vertices from smaller chains! Here we report a procedure which allows one to derive the generalized Wheland polynomial for a chain having an odd number of vertices. This is achieved by combining the coefficients for rings having the same number of vertices. The generalized Wheland polynomials for odd rings are simply related to the generalized Wheland polynomials for smaller chains and can be derived from the information on smaller chains. This makes it possible to extend the recursion for generalized Wheland polynomials for arbitrarily largen.