The isomorphism problem of trees from the viewpoint of Terwilliger algebras

TL;DR

This paper investigates how Terwilliger algebras can be used to determine the isomorphism class of finite rooted trees, providing algebraic tools for tree isomorphism problems.

Contribution

It demonstrates that the Terwilliger algebra associated with a rooted tree uniquely identifies its isomorphism class, advancing algebraic methods in graph isomorphism.

Findings

Terwilliger algebra recognizes the isomorphism class of rooted trees

Structure of the principal T-module is characterized

Algebraic approach to tree isomorphism problem

Abstract

Let $\Gamma^{(x_0)}$ be a finite rooted tree, for which $\Gamma$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $\Gamma$ with respect to $x_0$. We study the structure of the principal $T$-module. As a result, it is shown that $T$ recognizes the isomorphism class of $\Gamma^{(x_0)}$.