Terwilliger algebra of Odd graphs

TL;DR

This paper determines the Terwilliger algebra of the Odd graph, a specific distance graph related to Johnson schemes, extending previous work on Johnson schemes to this particular class of graphs.

Contribution

It explicitly computes the Terwilliger algebra of the Odd graph and provides its basis, filling a gap in algebraic combinatorics related to distance-regular graphs.

Findings

Explicit basis for the Terwilliger algebra of the Odd graph

Extension of Johnson scheme results to Odd graphs

Enhanced understanding of algebraic structure of Odd graphs

Abstract

In [The Terwilliger algebra of the Johnson schemes, Discrete Mathematics 307 (2007) 1621--1635], Levstein and Maldonado computed the Terwilliger algebra of the Johnson scheme $J(n,m)$ when $3m\leq n$. The distance-$m$ graph of $J(2m+1,m)$ is the Odd graph $O_{m+1}$. In this paper, we determine the Terwilliger algebra of $O_{m+1}$ and give its basis.