TY - JOUR

T1 - Recurrences for the genus polynomials of linear sequences of graphs

AU - Chen, Yichao

AU - Gross, Jonathan L.

AU - Mansour, Toufik

AU - Tucker, Thomas W.

N1 - 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.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - Given a finite graph H, the nth member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in Gn-1 by adding edges or identifying vertices, always in the same way. The genus polynomial ΓG(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 ΓGn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, ⋯, k with polynomials ΓGnj (z), for j = 1, 2, ⋯, k; from these polynomials, we form a column vector Vn(z)=[ΓGn1(z),ΓGn2(z), that satisfies a recursion Vn(z) = M(z)Vn-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 kth degree linear recursion for Γn(z), allowing us to avoid the partitioning, thereby yielding a reduction from k2 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.

AB - Given a finite graph H, the nth member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in Gn-1 by adding edges or identifying vertices, always in the same way. The genus polynomial ΓG(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 ΓGn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, ⋯, k with polynomials ΓGnj (z), for j = 1, 2, ⋯, k; from these polynomials, we form a column vector Vn(z)=[ΓGn1(z),ΓGn2(z), that satisfies a recursion Vn(z) = M(z)Vn-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 kth degree linear recursion for Γn(z), allowing us to avoid the partitioning, thereby yielding a reduction from k2 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.

KW - genus polynomial

KW - log-concavity

UR - http://www.scopus.com/inward/record.url?scp=85086263695&partnerID=8YFLogxK

U2 - https://doi.org/10.1515/ms-2017-0368

DO - https://doi.org/10.1515/ms-2017-0368

M3 - Article

SN - 0139-9918

VL - 70

SP - 505

EP - 526

JO - Mathematica Slovaca

JF - Mathematica Slovaca

IS - 3

ER -