Iterated claws have real-rooted genus polynomials

TL;DR

This paper proves that the genus polynomials of iterated claw graphs are real-rooted, advancing the understanding of their log-concavity and providing a new approach using matrix representations of graph operations.

Contribution

It establishes the real-rootedness of genus polynomials for iterated claws, extending previous work on graph operations and polynomial properties.

Findings

Genus polynomials of iterated claws are real-rooted.

Matrix representation aids in analyzing topological graph operations.

Supports the conjecture that all graph genus distributions are log-concave.

Abstract

We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a commonly observable condition on their partitioned genus distributions, even though it had been proved previously that iterative amalgamation does not always preserve real-rootedness of the genus polynomial of the iterated graph. In this paper, the iterated topological operations are adding a claw and adding a 3-cycle, rather than vertex- or edge-amalgamation. Our analysis here illustrates some advantages of employing a matrix representation of the transposition of a set of productions.