Log-concavity of the genus polynomials of Ringel Ladders

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>A </span><span>Ringel ladder </span><span>can be formed by a </span><span>self-bar-amalgamation </span><span>operation on a </span><span>symmetric ladder</span><span>, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which </span><span>linear chains </span><span>of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish </span><span>Ringel ladders </span><span>as the first significant </span><span>non-linear </span><span>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 </span><span>partitioned genus distribution </span><span>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. </span></p></div></div></div