Lattices and Norton algebras of Johnson, Grassmann and Hamming graphs

Abstract

To each of the Johnson, Grassmann and Hamming graphs we associate a lattice and characterize the eigenspaces of the adjacency operator in terms of this lattice . We also show that each level of the lattice induces in a natural way a tight frame for each eigenspace. For the most important eigenspace we compute explicitly the constant associated to the tight frame. Using the lattice we also give a formula for the product of the Norton algebra attached to that eigenspace.

Nonassociativity of the Norton Algebras of some distance regular graphs

“…In this section we summarize the results by Levstein, Maldonado and Penazzi [17] and Maldonado and Penazzi [20] on the Norton algebras of certain distance regular graphs, and extend the result from the hypercube graphs to all Hamming graphs. The reader is referred to Brouwer-Cohen-Neumaier [5] and van Dam-Koolen-Tanaka [7] for more background information on distance regular graphs.…”

“…Levstein-Maldonado-Penazzi [17] and Maldonado-Penazzi [20] showed the following result (whose proof in the hypercube case remains valid for all Hamming graphs).…”

“…To study the Norton algebras of J(n, k), Maldonado and Penazzi [20] constructed a lattice L which consists of all subsets of [n] with cardinality at most k together with 1 := [n]. The lattice L is ordered by containment with minimum element 0 := ∅ and maximum element 1.…”

“…The rank function of L is given by the dimension of linear spaces. The formulas for the meet and join of L are To study the Norton algebras of the Hamming graph H(d, e), we construct a lattice L which agrees with the lattice given by Maldonado and Penazzi [20] in the special case of e = 2.…”

“…See, e.g., Brouwer, Cohen and Neumaier [5, § 9.4] for more details. Levstein-Maldonado-Penazzi [17] and Maldonado-Penazzi [20] obtained formulas for the Norton product ⋆ on the eigenspace V 1 of Γ, where Γ is the Johnson graph J(n, k), the Grassman graph J q (n, k), the hypercube graph H(d, 2) or a dual polar graph. Using these formulas together with an extended formula that we obtain from H(d, 2) to the Hamming graph H(d, e), we determine the nonassociativity measurement C ⋆,n .…”