Low degree minimal generators of phylogenetic semigroups

TL;DR

This paper characterizes degree two minimal generators of phylogenetic semigroups on trivalent graphs, extending previous results to more complex graph structures with specific Betti numbers.

Contribution

It provides a complete description of degree two minimal generators for phylogenetic semigroups on all trivalent graphs with Betti numbers 1 and 2.

Findings

Degree two minimal generators characterized for all trivalent graphs with Betti number 1 and 2.

Extended understanding of minimal generating sets beyond previously studied cases.

Provides explicit descriptions for these generators in the context of phylogenetic semigroups.

Abstract

The phylogenetic semigroup on a graph generalizes the Jukes-Cantor binary model on a tree. Minimal generating sets of phylogenetic semigroups have been described for trivalent trees by Buczy\'nska and Wi\'sniewski, and for trivalent graphs with first Betti number 1 by Buczy\'nska. We characterize degree two minimal generators of the phylogenetic semigroup on any trivalent graph. Moreover, for any graph with first Betti number 1 and for any trivalent graph with first Betti number 2 we describe the minimal generating set of its phylogenetic semigroup.