Log-Concavity of the Grothendieck Classes of Banana Graphs and Clasped Necklaces

Abstract

The Grothendieck classes of melonic graphs satisfy a recursive relation and may be written as polynomials in the class of the moduli space ${\mathcal{M}}_{0,4}$ with nonnegative integer coefficients, conjectured to be log-concave. In this article, we investigate log-concavity and ultra-log-concavity for the Grothendieck class of banana graphs and the three families of polynomials involved in the recursive relation. We prove that all four are log-concave, establishing the specific case of banana graphs for the log-concavity conjecture. We additionally introduce the infinite family of clasped necklaces, melonic graphs obtained by replacing an edge of a $2$-banana with a string of $m$-bananas. Using the recursive relation, we explicitly compute the classes of clasped necklaces and prove that they too are log-concave.

29 pages, 2 figures, fixed some typos