Log-concavity of the genus polynomials of Ringel ladders
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.
Keywords
Cite
@article{arxiv.1501.06106,
title = {Log-concavity of the genus polynomials of Ringel ladders},
author = {J. L. Gross and T. Mansour and T. W. Tucker and D. G. L. Wang},
journal= {arXiv preprint arXiv:1501.06106},
year = {2015}
}
Comments
16 pages, 6 figures