The Tutte polynomials of catacondensed benzenoid systems

A catacondensed benzenoid system (resp. benzenoid chain) is a benzenoid system whose inner dual graph is a tree (resp. a path). The Tutte polynomial of a graph is a two-variable polynomial whose evaluations at various points are equivalent to the exact solutions of many counting problems. In this paper, we introduce a graph vector at a given edge which related to the Tutte polynomial. Based on this concept and by three classes transfer matrices, we get the reduction formula for Tutte polynomial of any catacondensed benzenoid system. Moreover, the number of spanning trees for any catacondensed benzenoid system is also determined via a product of (2×2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2\times 2)$$\end{document} matrices with entries in N. As a by-product, we study the extremum problem of the number of spanning trees over the set of cataconsed hexagonal systems with one branched hexagon.