The Terwilliger algebras of Odd graphs and Doubled Odd graphs

For an integer m > 1, let S = {1, 2, ..., 2m + 1}. }. Denote by 2.Om+1 the Doubled Odd graph on S with vertex set X := ((S)(m)) boolean OR ( (S) (m + 1)). By folding this graph, one can obtain a new graph called Odd graph Om + 1 with vertex set K := ((S)(m)). In this paper, we shall study the m Terwilliger algebras of 2.Om+1. Om + 1 and Om + 1 . We first consider the case of Om+1. With respect to any fixed vertex x(0) is an element of X, let A := A(x(0)) denote the centralizer algebra of the stabilizer of x(0) in the automorphism group of Om+1, and .J := .J (x(0) ) the Terwilliger algebra of Om + 1. For the algebras d and .J : (i) we construct a basis of d by the stabilizer of x(0) acting on X x X, compute its dimension and show that d = .J ; (ii) for m >= 3, we give all the isomorphism classes of irreducible .J- modules and display the decomposition of .J in a block-diagonal form (up to isomorphism). These results together with the relations between 2.Om+1 and Om+1 allow us to further study the corresponding centralizer algebra and Terwilliger algebra for 2.Om+1. Consequently, the results in the above (i), (ii) for2.Om+1 can be similarly generalized to the case of 2.Om+1 moreover, we define three subalgebras of the Terwilliger algebra of Om+1 such that their direct sum is just this algebra. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.