On Strongly Closed Subgraphs with Diameter Two and Q-Polynomial Property∗ (Preliminary Version 2.0.0)
نویسنده
چکیده
In this paper, we study a distance-regular graph Γ = (X,R) with an intersection number a2 6= 0 having a strongly closed subgraph Y of diameter 2. Let E0, E1, . . . , ED be the primitive idempotents corresponding to the eigenvalues θ0 > θ1 > · · · > θD of Γ. Let V = C be the vector space consisting of column vectors whose rows are labeled by the vertex set X. Let W be the subspace of V consisting of vectors whose supports lie in Y . A nonzero vector v ∈W is said to be tight if E0v = Eiv = 0 for some i = 1, 2, . . . , D. We show that the existence of a tight vector in W is equivalent to a balanced condition defined by P. Terwilliger. As an application, we study the structure of parallelogram-free distance-regular graphs and conditions for these graphs to be Q-polynomial.
منابع مشابه
On strongly closed subgraphs with diameter two and the Q-polynomial property
1Introduction Let $\Gamma=(X, R)$ be a distance-regular graph (DRG) of diameter $\Gamma_{j}(u)=\{x\in X|\partial(u, x)=j\}$ and $\Gamma(u)=\Gamma_{1}(u)$. For two vertices $u$ and $v\in X$ with $\partial(u, v)=j$ let $C(u., v)$ $=$ $\Gamma_{j-1}(u)\cap\Gamma(v)$ , $A(u, v)$ $=$ $\Gamma_{j}(u)\cap\Gamma(v)$ , and $B(u, v)$ $=$ $\Gamma_{j+1}(u)\cap\Gamma(v)$ .
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