On the graph labellings arising from phylogenetics

TL;DR

This paper investigates semigroups of graph labellings related to phylogenetics, establishing bounds on generator degrees and providing examples where these bounds are tight.

Contribution

It introduces bounds on the degrees of generators of semigroups associated with graphs in phylogenetics, extending previous models and providing sharp examples.

Findings

Bound on generator degrees by g+1 for graphs with Betti number g

Semigroups generalize the Jukes-Cantor model and phylogenetic toric varieties

Examples demonstrate the sharpness of the bounds

Abstract

We study semigroups of labellings associated to a graph. These generalize the Jukes-Cantor model and phylogenetic toric varieties defined by Buczy\'nska. Our main theorem bounds the degree of the generators of the semigroup by g+1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.