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

## TL;DR

This paper investigates the algebraic structure of ages of certain countable metrically homogeneous graphs, showing that under specific conditions, their age algebras are polynomial, often in infinitely many variables.

## Contribution

It extends Cameron's results to metrically homogeneous graphs of generic type, demonstrating their age algebras are polynomial, often with infinitely many variables.

## Key findings

- Age algebras of certain metrically homogeneous graphs are polynomial.
- These algebras are typically in infinitely many variables.
- The results generalize Cameron’s conditions to a broader class of graphs.

## Abstract

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.