## Abstract

Given a finite graph H, the n^{th} member G_{n} of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G_{n-1} by adding edges or identifying vertices, always in the same way. The genus polynomial Γ_{G}(z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ_{G}_{n}(z) for the graphs in a linear family have been obtained by partitioning the embeddings of G_{n} into types 1, 2, ⋯, k with polynomials ΓG_{n}^{j} (z), for j = 1, 2, ⋯, k; from these polynomials, we form a column vector V_{n}(z)=[ΓG_{n}^{1}(z),ΓG_{n}^{2}(z), that satisfies a recursion V_{n}(z) = M(z)V_{n-1}(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a k^{th} degree linear recursion for Γ_{n}(z), allowing us to avoid the partitioning, thereby yielding a reduction from k^{2} multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Original language | English |
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Pages (from-to) | 505-526 |

Number of pages | 22 |

Journal | Mathematica Slovaca |

Volume | 70 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jun 2020 |

### Bibliographical note

Funding Information:T. W. Tucker is supported by Simons Foundation Grant #317689.

Funding Information:

J. L. Gross is supported by Simons Foundation Grant #315001.

Funding Information:

Yichao Chen is supported by the NNSFC under Grant No. 11471106.

Publisher Copyright:

© 2020 Mathematical Institute Slovak Academy of Sciences 2020.

## Keywords

- genus polynomial
- log-concavity

## ASJC Scopus subject areas

- Mathematics (all)