Tropicalizing the Graph Profile of Some Almost-Stars

. 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 \ (U\), the tropicalization of the graph profile of \ (U\) essentially records all valid pure binomial inequalities involving graph homomorphism numbers for graphs in \ (U\). Building upon ideas and techniques described by Blekherman and Raymond in 2022, we compute the tropicalization of the graph profile for the graph containing a single vertex as well as stars where one edge is subdivided. 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 graph involved. Keywordsgraph profilegraph homomorphismtropicalizationtreesstarsMSC codes05C3590C3514T90