On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0

Abstract

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ, and let A i (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A 1. For x ∈ X and for 0 ≤ i ≤ D, let Γ i (x) denote the set of vertices in X that are distance i from vertex x. Fix x ∈ X and let T = T (x) denote the subalgebra of Mat X (C) generated by

On (almost) $2$-$Y$-homogeneous distance-biregular graphs

“…The main ideas for both defining the scalar ∆ i and proving some related results are taken from the theory of bipartite distance-regular graphs. See for example [5,27,28,33,40,41] for more details. The next inequality will be important and useful later.…”

Certain graphs with exactly one irreducible $T$-module with endpoint $1$, which is thin: the pseudo-distance-regularized case

“…There has been a sizeable amount of research investigating (distance-regular) graphs that have a Terwilliger algebra T with, up to isomorphism, just a few T -modules of a certain endpoint, all of which are (non-)thin (with respect to a certain base vertex); see for example [5,6,7,8,16,17,18,19,20,21,22,23]. These studies generally try to show that such algebraic conditions hold if and only if certain combinatorial conditions are satisfied.…”