Forcing Polynomial of Double Hexagonal Chains

Abstract

For hexagonal systems, the maximum forcing number is equal to the Clar number, which is an invariant to estimate the resonance energy. The forcing number of a perfect matching M of a graph G was introduced by Harary et al., which is the smallest cardinality over all subsets of M that are not included by any other perfect matchings of G. The same concept under the name ‘degree of freedom’ was proposed by Klein and Randić in earlier time. The first author and collaborators introduced the forcing polynomial of a graph as an enumerating polynomial for perfect matchings with the same forcing number. In this paper, we derive a recurrence relation of forcing polynomial for double hexagonal chains, which is a hexagonal system constructed by successive triple-edge fusions of naphthalenes. As consequences, we obtain explicit forms of forcing polynomials for double linear and zigzag hexagonal chains, and some special examples.