The Terwilliger algebra of the doubled Odd graph

TL;DR

This paper investigates the algebraic structure of the doubled Odd graph, providing a basis for its centralizer algebra, classifying its irreducible modules, and showing it coincides with the Terwilliger algebra, revealing new properties of this bipartite graph.

Contribution

It establishes the equality of the centralizer algebra and the Terwilliger algebra for the doubled Odd graph, a novel example in bipartite non-Q-polynomial distance-transitive graphs.

Findings

Basis for the centralizer algebra $\\mathcal{A}$ determined

Irreducible $T$-modules classified for $m\geq 3$

Centralizer algebra and Terwilliger algebra shown to be equal

Abstract

Let $2.O_{m+1}$ denote the doubled Odd graph with vertex set $X$ on a set of cardinality $2m+1$, where $m\geq 1$. Fix a vertex $x_0\in X$. Let $\mathcal{A}:=\mathcal{A}(x_0)$ denote the centralizer algebra of the stabilizer of $x_0$ in the automorphism group of $2.O_{m+1}$, and $T:=T(x_0)$ the Terwilliger algebra of $2.O_{m+1}$. In this paper, we first give a basis of $\mathcal{A}$ by considering the action of the stabilizer of $x_0$ on $X\times X$ and determine the dimension of $\mathcal{A}$. Furthermore, we give three subalgebras of $\mathcal{A}$ such that their direct sum is $\mathcal{A}$ as vector space. Next, for $m\geq 3$ we find all isomorphism classes of irreducible $T$-modules to display the decomposition of $T$ in a block-diagonalization form. Finally, we show that the two algebras $\mathcal{A}$ and $T$ coincide. This result tells us that the graph $2.O_{m+1}$ may be the first example of bipartite but not $Q$-polynomial distance-transitive graph for which the corresponding centralizer algebra and Terwilliger algebra are equal.