Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

TL;DR

This paper establishes conditions under which linear combinations of log-concave sequences remain log-concave, and applies these results to prove the log-concavity of genus distributions in certain graphs, supporting a long-standing conjecture.

Contribution

It introduces new conditions called synchronicity and ratio-dominance that ensure the preservation of log-concavity in combined sequences and applies these to genus distributions of graphs.

Findings

Proved log-concavity of genus distributions for graphs built from specific fragments.

Identified conditions that guarantee log-concavity in linear combinations of sequences.

Demonstrated preservation of log-concavity through iterative graph constructions.

Abstract

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called \textit{synchronicity} and \textit{ratio-dominance}, and a characterization of some bivariate sequences as \textit{lexicographic}. We are motivated by the 25-year old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the \textit{partitioned genus distributions} of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.