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 for any integer i, let Γi(x) denote the set of vertices at distance i from x. ...Let V=ℂX 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∈X let ẑ 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∈Γ2(x) and v∈Γ2(x)∩Γ2(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 ẑ, 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.
Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D$. In Caughman (2004), Caughman showed that if $D \ge 12$, then $\Gamma$ is $Q$-polynomial if and only if one of the following ...(i)-(iv) holds: (i) $\Gamma$ is the ordinary $2D$-cycle, (ii) $\Gamma$ is the Hamming cube $H(D,2)$, (iii) $\Gamma$ is the antipodal quotient of the Hamming cube $H(2D,2)$, (iv) the intersection numbers of $\Gamma$ satisfy $c_i = (q^i - 1)/(q-1)$, $b_i = (q^D-q^i)/(q-1)$ $(0 \le i \le D)$, where $q$ is an integer at least $2$. In this paper we show that the above result is true also for bipartite distance-regular graphs with $D \in \{9,10,11\}$.
A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial ...and algebraic) proofs of the fact that every edge-distance-regular graph Γ is distance-regular and homogeneous. More precisely, Γ is edge-distance-regular if and only if it is bipartite distance-regular or a generalized odd graph. Also, we obtain the relationships between some of their corresponding parameters, mainly, the distance polynomials and the intersection numbers.
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 denote the adjacency matrix of Γ. For x∈X let T=T(x) denote the ...subalgebra of MatX(C) generated by A, E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim Ei⁎W≤1 for 0≤i≤D. By the endpoint of W we mean min{i|Ei⁎W≠0}. For 0≤i≤D, let Γi(z) denote the set of vertices in X that are distance i from vertex z. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0 and for 2≤i≤D−2 there exist complex scalars αi,βi with the following property: for all x,y,z∈X such that ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|; (ii) For all x∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x) with endpoint two, and these modules are thin.
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