The isomorphism problem of trees from the viewpoint of Terwilliger algebras

Let ${\mathrm{\Gamma }}^{(x_0)}$ be a finite rooted tree, for which $\mathrm{\Gamma }$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $\mathrm{\Gamma }$ 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 ${\mathrm{\Gamma }}^{(x_0)}$.