The Algebra of an Age for Metrically Homogeneous Graphs of Generic Type

Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. Here we consider a class of countable metrically homogeneous graphs. The algebra of an age is a concept introduced by Cameron and is closely connected to the profile of the automorphism group of the associated countable structure. Cameron later provided sufficient structural conditions on the age of $\aleph_0$-categorical countable homogeneous structures for showing that the algebra of the age is a polynomial algebra. In this paper, we use Cameron's result to deduce that the algebra of the age of certain metrically homogeneous graphs of generic type are polynomial algebras, typically in infinitely many variables.