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arXiv · 1501.06106

Log-concavity of the genus polynomials of Ringel ladders

J. L. Gross·T. Mansour·T. W. Tucker·D. G. L. Wang
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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.

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J. L. Gross, T. Mansour, T. W. Tucker, D. G. L. Wang. 2015-01-25. Log-concavity of the genus polynomials of Ringel ladders. https://arxiv.org/abs/1501.06106

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