The isomorphism problem of trees from the viewpoint of Terwilliger algebras with respect to an edge

Abstract

Let Γ be a finite connected tree, and let $x_0{x}_0^{\prime }$ be an edge of Γ. Fix $X_0=\{x_0,x_0^{\prime }\}$ as the base vertex set, let ${\Gamma }^{\left(X_0\right)}$ be an edge rooted tree with the edge root $X_0$. Let $T=T\left(X_0\right)$ be the Terwilliger algebra of ${\Gamma }^{\left(X_0\right)}$ with respect to $X_0$. In this paper, we characterize the structure of the irreducible T-module with the endpoint 0. As a result, it is shown that $T=T\left(X_0\right)$ recognizes the isomorphism class of ${\Gamma }^{\left(X_0\right)}$.