Genus polynomials of ladder-like sequences of graphs

Production matrices have become established as a general paradigm for calculating the genus polynomials for linear sequences of graphs. Here we derive a formula for the production matrix of any of the linear sequences of graphs that we call ladder-like, where any connected graph H with two 1-valent root vertices may serve as a super-rung throughout the ladder. Our main theorem expresses the production matrix for any ladder-like sequence as a linear combination of two fixed 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document} matrices, taken over the ring of polynomials with integer coefficients. This leads to a formula for the genus polynomials of the graphs in the ladder-like sequence, based on the two partial genus polynomials of the super-rung. We give a closed formula for these genus polynomials, for the case in which all imbeddings of the super-rung H are planar. We show that when the super-rung H has Betti number at most one, all the genus polynomials in the sequence are log-concave.