منابع مشابه
On bipartite Q-polynomial distance-regular graphs
Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, diameter d ≥ 3 and valency k ≥ 3. Let RX denote the vector space over R consisting of column vectors with entries in R and rows indexed by X. For z ∈ X, let ẑ denote the vector in RX with a 1 in the z-coordinate, and 0 in all other coordinates. Fix x, y ∈ X such that ∂(x, y) = 2, where ∂ denotes path-length distance...
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Let denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, . . . , θD is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence θi−1 − βθi + θi+1 = 0 (0 < i < D), for some real scalar β. Let q denote a complex scalar such that q + q−1 = β. Bannai and Ito have conjectured that q is real if the diameter D is suffi...
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Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy ai = 0 for 0 ≤ i ≤ D − 1 and aD 6= 0. We show that Γ is a polygon, a folded cube, or an Odd graph.
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We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ≥ 3. Let E denote a minimal idempotent of Γ which is not the trivial idempotent E0. Let {θ∗ i }i=0 denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ◦ E is a linear combination of E0, E, and at...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1996
ISSN: 0095-8956
DOI: 10.1006/jctb.1996.0061