Log-concavity of the genus polynomials of Ringel ladders

TL;DR

This paper proves that Ringel ladders, a specific class of graphs formed by self-bar-amalgamation, have log-concave genus polynomials, supporting the broader conjecture that all graphs possess this property.

Contribution

It introduces the first significant non-linear infinite family of graphs with log-concave genus polynomials and develops an algebraic matrix approach using Chebyshev polynomials.

Findings

Ringel ladders have log-concave genus polynomials.

Established a matrix-based algebraic representation of self-bar-amalgamation.

Used Chebyshev polynomials to analyze the genus polynomial.

Abstract

A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polynomials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.