## Abstract

The genus polynomial for a finite graph G is the generating function gG(z) = Σaiz

i, where ai is the number of imbeddings of G in the surface of genus i. A linear family Gn of graphs is formed by taking n copies of the same graph G and forming a path of them by adding edges in the same way between one copy of G and the next. For any such linear family there is a production or transfer matrix M(z) and initial vector v(z) (all entries are polynomials in z with non-negative integer coefficients) such that the genus polynomials for the imbedding types of Gn are given by Mn (z)v(z). The columns of M(1) have constant column sum s so (1/s)M(1) is a matrix for a Markov chain whose states are the imbedding types

of the linear family. We show how to use the Jordan normal form for M(1) to find the average genus of each imbedding type for each member of a linear family.

i, where ai is the number of imbeddings of G in the surface of genus i. A linear family Gn of graphs is formed by taking n copies of the same graph G and forming a path of them by adding edges in the same way between one copy of G and the next. For any such linear family there is a production or transfer matrix M(z) and initial vector v(z) (all entries are polynomials in z with non-negative integer coefficients) such that the genus polynomials for the imbedding types of Gn are given by Mn (z)v(z). The columns of M(1) have constant column sum s so (1/s)M(1) is a matrix for a Markov chain whose states are the imbedding types

of the linear family. We show how to use the Jordan normal form for M(1) to find the average genus of each imbedding type for each member of a linear family.

Original language | English |
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Title of host publication | AMS National Meeting |

State | Published - 2018 |