Genus polynomials of ladder-like sequences of graphs

Yichao Chen, Jonathan L. Gross, Toufik Mansour, Thomas W. Tucker

Research output: Contribution to journalArticlepeer-review


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 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.

Original languageEnglish
Pages (from-to)137-155
Number of pages19
JournalJournal of Algebraic Combinatorics
Issue number2
StatePublished - 1 Sep 2020

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.


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

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


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