Gutman trees. Combinatorial–recursive relations of counting polynomials: data reduction using chemical graphs

Abstract

When m_i vertices of degree one are added to the ith vertex of a path graph on n vertices, P_n, a Gutman tree, P_n(m₁, m₂, …, m_n), results. It is demonstrated that such trees unify theorems of several counting polynomials, including the counting polynomial of Hosoya, the resonance polynomial of Aihara, and matching, characteristic, sextet and independence polynomials. A ‘polygraphic sequence’ of integers identifies a polyhex graph, a Clar graph and a tree graph. Further, such trees introduce a novel and purely graph-theoretical means of data reduction, whereby topological properties of non-branched and several types of branched catacondensed and certain pericondensed benzenoid hydrocarbons may be efficiently ‘stored’ in these trees. Recursive relations of periodic families of Gutman trees generate models of benzenoid hydrocarbons satisfying the conjecture of Aihara and thus represents an area where the simple Clar sextet formalism and ‘Dewartype’ resonance theory coincide. Further, it is shown how such recursions derive the method of secular-determinant reduction elaborated by Balasubramanian for obtaining spectra of trees. Several combinatorial–recursive relations are derived which demonstrate the topological dependence of Kekulé counts and are of value in the problem of graph recognition.