Log-concavity of the genus polynomials of Ringel Ladders

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. 1. Genus Polynomials Our graphs are implicitly taken to be connected, and our graph embeddings are cellular and orientable. For general background in topological graph theory, see [13, 1]. Prior acquaintance with the concepts of partitioned genus distribution (abbreviated here as pgd) and production (e.g., [10, 17]) are necessary preparation for reading this paper. The exposition here is otherwise intended to be accessible both to graph theorists and to combinatorialists. The number of combinatorially distinct embeddings of a graph G in the orientable surface of genus i is denoted by gi(G). The sequence g0(G), g1(G), g2(G), . . ., is called the genus distribution of G. A genus distribution contains only finitely many positive numbers, and there are no zeros between the first and last positive numbers. The genus polynomial is the polynomial ΓG(x) = g0(G) + g1(G)x + g2(G)x 2 + . . . . Log-concave sequences A sequence A = (ak) n k=0 is said to be nonnegative, if ak ≥ 0 for all k. An element ak is said to be an internal zero of A if ak = 0 and if there exist indices i and j with i < k < j, such that aiaj 6= 0. If ak−1ak+1 ≤ ak for all k, then A is 2000 Mathematics Subject Classification. 05A15, 05A20, 05C10. 1Supported by Simons Foundation Grant #315001. 2Supported by Simons Foundation Grant #317689. 3Supported by Beijing Institute of Technology Research Fund Program for Young Scholars.