A characterization of Q-polynomial distance-regular graphs

Pascasio, Arlene A.

doi:10.1016/j.disc.2007.08.034

Cited by 10 publications

(15 citation statements)

References 3 publications

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“…Our next general goal is to determine whether the equations (23) give a uniform or strongly uniform structure. In order to do this we introduce some parameters q and s * .…”

Section: Uniform Structures On a Posetmentioning

confidence: 99%

“…In his thesis [12] Delsarte introduced the Q-polynomial property for a distance-regular graph Γ (see Section 2 for formal definitions). Since then the Q-polynomial property has been investigated by many authors, such as Bannai and Ito [1], Brouwer, Cohen and Neumaier [3], Caughman [4,5,6,7,8,9], Curtin [10,11], Jurišić, Terwilliger, and Žitnik [14], Lang [15,16], Lang and Terwilliger [17], Miklavič [18,19,20,21], Pascasio [22,23], Tanaka [24,25], Terwilliger [26,27,30,32], and Weng [33,34].…”

Section: Introductionmentioning

confidence: 99%

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Bipartite Q-polynomial distance-regular graphs and uniform posets

Miklavič

Terwilliger

2012

J Algebr Comb

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Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D ≥ 3. Fix x ∈ X and let L (resp. R) denote the corresponding lowering (resp. raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL 2 , LRL, L 2 R, L. Define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

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“…Our next general goal is to determine whether the equations (23) give a uniform or strongly uniform structure. In order to do this we introduce some parameters q and s * .…”

Section: Uniform Structures On a Posetmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bipartite Q-polynomial distance-regular graphs and uniform posets

Miklavič

Terwilliger

2012

J Algebr Comb

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“…Theorem 1.1 gives a characterization of the Q -polynomial distance-regular graphs. A similar characterization, where assumption (i) is replaced by some equations involving the dual eigenvalues and intersection numbers, is given by Pascasio [7].…”

Section: Introductionmentioning

confidence: 97%

The Q-polynomial idempotents of a distance-regular graph

Jurišić

Terwilliger

itnik

2010

Journal of Combinatorial Theory, Series B

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We obtain the following characterization of Q -polynomial distanceregular graphs. Let Γ denote a distance-regular graph with diam-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 E 0 , E, and at most one other minimal idempotent of Γ ; (ii) there exists a complex scalar β such that θ * i−1 − βθ * i + θ * i+1 is independent of i for 1 i d − 1; (iii) θ * i = θ * 0 for 1 i d.

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“…There is a characterization involving the dual distance matrices [10,Theorem 3.3]. There is a characterization involving the intersection numbers a i [8,Theorem 3.8]; cf. [3,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%