A combinatorial basis for Terwilliger algebra modules of a bipartite distance-regular graph

Let Gamma denote a bipartite distance-regular graph with diameter D >= 4 and valency k >= 3. Let X denote the vertex set of Gamma, and for any integer i, let Gamma(i)(x) denote the set of vertices at distance i from x. Let V = C-x denote the vector space over C consisting of column vectors whose coordinates are indexed by X and whose entries are in C, and for z is an element of X let (z) over cap denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x, u, v where u is an element of Gamma(2)(x) and v is an element of Gamma(2)(x) boolean AND Gamma(2)(u), and let T = T(x) denote the Terwilliger algebra with respect to x. Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T-module of endpoint 2, and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors (z) over cap, where the vertices z are chosen based on the their distances from x, u, and v. We use this T-module to construct combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman-Singleton graph, Tutte's 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The paper discusses bipartite distance-regular graphs with diameter at least 4 and valency at least 3, constructing combinatorially-defined spanning sets for T-modules of endpoint 2 with certain assumptions, and examining the action of the adjacency matrix on these sets. By using this T-module, combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 are constructed for various graph examples. Additionally, a list of several other graphs satisfying the conditions is provided.