The Terwilliger algebras of Odd graphs and Doubled Odd graphs

For an integer $m\ge 1$, let $S=\{1,2,\dots ,2m+1\}$. Denote by $2.O_{m+1}$ the Doubled Odd graph on S with vertex set $X:=\left(\begin{array}{c}S\\ m\end{array}\right)\cup \left(\begin{array}{c}S\\ m+1\end{array}\right)$. By folding this graph, one can obtain a new graph called Odd graph $O_{m+1}$ with vertex set $X:=\left(\begin{array}{c}S\\ m\end{array}\right)$. In this paper, we shall study the Terwilliger algebras of $2.O_{m+1}$ and $O_{m+1}$. We first consider the case of $O_{m+1}$. With respect to any fixed vertex $x_0\in X$, let $A:=A\left(x_0\right)$ denote the centralizer algebra of the stabilizer of $x_0$ in the automorphism group of $O_{m+1}$, and $T:=T\left(x_0\right)$ the Terwilliger algebra of $O_{m+1}$. For the algebras $A$ and $T$: (i) we construct a basis of $A$ by the stabilizer of $x_0$ acting on $X\times X$, compute its dimension and show that $A=T$; (ii) for $m\ge 3$, we give all the isomorphism classes of irreducible $T$-modules and display the decomposition of $T$ in a block-diagonal form (up to isomorphism). These results together with the relations between $2.O_{m+1}$ and $O_{m+1}$ allow us to further study the corresponding centralizer algebra and Terwilliger algebra for $2.O_{m+1}$. Consequently, the results in the above (i), (ii) for $O_{m+1}$ can be similarly generalized to the case of $2.O_{m+1}$; moreover, we define three subalgebras of the Terwilliger algebra of $2.O_{m+1}$ such that their direct sum is just this algebra.