Coulson function and Hosoya index: extension of the relationship to polycyclic graphs and to new types of matching polynomials

Abstract

Gutman et al. [Chem. Phys. Lett. 355 (2002) 378–382] established a relationship between the Coulson function, \(F(G,x) \,{=}\,n - [\hbox {i}x{\phi}^{\prime}(G,\hbox {i}x)/{\phi}(G,\hbox{i}x)]\), where \phi is the characteristic polynomial, and the Hosoya index Z, which is the sum over all k of the counts of all k-matchings. Like the original Coulson function, this relationship was postulated only for trees. The present study shows that the relationship can be extended to (poly)cyclic graphs by substituting the matching, or acyclic, polynomial for the characteristic polynomial. In addition, the relationship is extended to new types of matching polynomials that match cycles larger than edges (2-cyc1es). Finally, this presentation demonstrates a rigorous mathematical relationship between the graph adjacency matrices and the coefficients of these polynomials and describes computational algorithms for calculating them.