Tropicalizing the Graph Profile of Some Almost-Stars

TL;DR

This paper computes the tropicalization of the graph profile for certain almost-star graphs, enabling the verification of binomial inequalities in graph homomorphism numbers via explicit linear programming.

Contribution

It extends the tropicalization approach to almost-star graphs, providing a method to verify binomial inequalities through linear programming.

Findings

Tropicalization computed for $K_1$ and $S_{2,1^k}$-trees.

Verification of inequalities reduces to linear programming.

Explicit method for almost-star graphs in extremal combinatorics.

Abstract

Many important problems in extremal combinatorics can be stated as certifying polynomial inequalities in graph homomorphism numbers, and in particular, many ask to certify pure binomial inequalities. For a fixed collection of graphs $\mathcal{U}$, the tropicalization of the graph profile of $\mathcal{U}$ essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in $\mathcal{U}$. Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for $K_1$ and $S_{2,1^k}$-trees, almost-star graphs with one branch containing two edges and $k$ branches containing one edge. This allows pure binomial inequalities in homomorphism numbers (or densities) for these graphs to be verified through an explicit linear program where the number of variables is equal to the number of edges in the biggest $S_{2,1^k}$-tree involved.