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Genus polynomials of ladder-like sequences of graphs

  • Yichao ChenEmail author
  • Jonathan L. Gross
  • Toufik Mansour
  • Thomas W. Tucker
Article
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Abstract

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 \(3\times 3\) 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.

Keywords

Linear sequences of graphs String operations Imbedding types Genus polynomials Partial genus polynomials Production matrices 

Notes

Supplementary material

10801_2019_897_MOESM1_ESM.pdf (308 kb)
Supplementary material 1 (pdf 308 KB)

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsSuZhou University of Science and TechnolgySuZhouChina
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsUniversity of HaifaHaifaIsrael
  4. 4.Department of MathematicsColgate UniversityHamiltonUSA

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