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Dual Bipartite Q-polynomial Distance-regular Graphs

1996
*
European journal of combinatorics (Print)
*

Let ⌫ denote a

doi:10.1006/eujc.1996.0052
fatcat:ydkjuzmyyfcardrctoamjozayi
*distance*-*regular**graph*with diameter d у 3 which is not*bipartite*. ... Curtin shows in [3] that a*bipartite**distance*-*regular**graph*of diameter d у 3 has a dual*bipartite**Q*-*polynomial*structure if f it is 2-homogeneous . Nomura shows in P ROOF . ...##
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On bipartite Q-polynomial distance-regular graphs

2007
*
European journal of combinatorics (Print)
*

Let Γ denote a

doi:10.1016/j.ejc.2005.09.003
fatcat:7ge2yrvpwbbnnbzn2sqpcpnvl4
*bipartite**Q*-*polynomial**distance*-*regular**graph*with vertex set X , diameter d ≥ 3 and valency k ≥ 3. ... We obtain our main result using Terwilliger's "balanced set" characterization of the*Q*-*polynomial*property. ... Introduction This paper is part of an ongoing effort to understand and classify the*Q*-*polynomial**bipartite**distance*-*regular**graphs*[3] [4] [5] [6] [7] [8] . ...##
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Bipartite Q-polynomial distance-regular graphs and uniform posets
[article]

2011
*
arXiv
*
pre-print

Let denote a

arXiv:1108.2484v1
fatcat:tfxrnwpntzfqdmze63wbrrsysi
*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 yields a certain linear dependency among RL^2, LRL, L^2R, L. Define a partial order < on X as follows. ...*Bipartite**distance*-*regular**graphs*We continue to discuss the*distance*-*regular**graph*Γ from Section 2. Recall that Γ is*bipartite*whenever a i = 0 for 0 ≤ i ≤ D. ...##
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Almost-bipartite distance-regular graphs with the Q-polynomial property
[article]

2005
*
arXiv
*
pre-print

Let G denote a

arXiv:math/0508435v1
fatcat:jpu4p4qx6rb6lag2jdgbk3j4xy
*Q*-*polynomial**distance*-*regular**graph*with diameter D at least 4. Assume that the intersection numbers of G satisfy a_i=0 for 0 ≤ i ≤ D-1 and a_D≠ 0. ... We show that G is a polygon, a folded cube, or an Odd*graph*. ... Lemma 3.1 Let Γ denote an almost-*bipartite**distance*-*regular**graph*with diameter D ≥ 3. Suppose that Γ is*Q*-*polynomial*but not as in Theorem 1.1(i)-(iii). Then Γ has a unique*Q*-*polynomial*eigenvalue. ...##
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Bipartite Q-Polynomial Quotients of Antipodal Distance-Regular Graphs

1999
*
Journal of combinatorial theory. Series B (Print)
*

Let 1 denote a

doi:10.1006/jctb.1999.1911
fatcat:f2sbbfywkjgbbhcboj2zy5d7yq
*bipartite**Q*-*polynomial**distance*-*regular**graph*with diameter D 4. We show that 1 is the quotient of an antipodal*distance*-*regular**graph*if and only if one of the following holds. ... Let 1 denote a*bipartite**distance*-*regular**graph*with diameter D 3 and valency k 3. Suppose 1 is*Q*-*polynomial*with respect to some primitive idempotent E. ... Let 1 denote a*bipartite**Q*-*polynomial**distance*-*regular**graph*with diameter D 4. Then 1 is the quotient of an antipodal distanceregular*graph*if and only if one of the following holds. ...##
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Bipartite Q-polynomial distance-regular graphs and uniform posets

2012
*
Journal of Algebraic Combinatorics
*

Let Γ denote a

doi:10.1007/s10801-012-0401-1
fatcat:dwea5vh66bgm3n6hw4xb3acwki
*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. ...*Bipartite**distance*-*regular**graphs*We continue to discuss the*distance*-*regular**graph*Γ from Sect. 2. Recall that Γ is*bipartite*whenever a i = 0 for 0 ≤ i ≤ D. ...##
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Almost-bipartite distance-regular graphs with the Q-polynomial property

2007
*
European journal of combinatorics (Print)
*

Let Γ denote a

doi:10.1016/j.ejc.2005.07.004
fatcat:mxm27cc6xbg5le53gutydzmkja
*Q*-*polynomial**distance*-*regular**graph*with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy a i = 0 for 0 ≤ i ≤ D − 1 and a D = 0. ... We show that Γ is a polygon, a folded cube, or an Odd*graph*. ... The*graph*2.Γ is*bipartite*and*distance*-*regular*with diameter 2D + 1. ...##
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On bipartite Q-polynomial distance-regular graphs with c2=1

2007
*
Discrete Mathematics
*

Let denote a

doi:10.1016/j.disc.2005.09.044
fatcat:vephvt5isrh4hpt27ngogaor24
*bipartite**Q*-*polynomial**distance*-*regular**graph*with diameter d 3, valency k 3 and intersection number c 2 = 1. ...*Q*-*polynomial*property Let denote a*distance*-*regular**graph*with diameter d 3. ... The classification of almost*bipartite**Q*-*polynomial**distance*-*regular**graphs*with girth 6 is given in [6] , so it remains to consider the case in which is*bipartite*. ...##
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The last subconstituent of a bipartite Q-polynomial distance-regular graph

2003
*
European journal of combinatorics (Print)
*

Let Γ denote a

doi:10.1016/s0195-6698(03)00059-3
fatcat:kxrdgvurivd3lccs227yzw7iry
*bipartite**distance*-*regular**graph*with diameter D ≥ 3. ... In this paper, we assume Γ is*Q*-*polynomial*and show Γ 2 D is*distance*-*regular*and*Q*-*polynomial*. We compute the intersection numbers of Γ 2 D from the intersection numbers of Γ . ... The*graph*Γ 2 D is*distance*-*regular*when Γ is*Q*-*polynomial*Definition 9.1. Let Γ denote a*bipartite**distance*-*regular**graph*with vertex set X and diameter D ≥ 3. ...##
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On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

2018
*
Electronic Journal of Combinatorics
*

Let $\Gamma$ denote a

doi:10.37236/7347
fatcat:scmlwl37wfbppg7un5qgmzihmi
*bipartite**distance*-*regular**graph*with diameter $D$. ... In this paper we show that the above result is true also for*bipartite**distance*-*regular**graphs*with $D \in \{9,10,11\}$. ... One such subclass is the class of*Q*-*polynomial**bipartite**distance*-*regular**graphs*. ...##
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On bipartite Q-polynomial distance-regular graphs with c 2 2

unpublished

Let Γ denote a

fatcat:5dxqex472bh2vlmcrbyhk2yaji
*bipartite**Q*-*polynomial**distance*-*regular**graph*with diameter D 4, valency k 3 and intersection number c 2 2. ... Let Γ denote a*Q*-*polynomial**bipartite**distance*-*regular**graph*with diameter D 6 and valency k 3. ... Let Γ denote a*bipartite**Q*-*polynomial**distance*-*regular**graph*with diameter D 4, valency k 3, and intersection number c 2 2. ...##
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The last subconstituent of the Hemmeter graph

2008
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Discrete Mathematics
*

We prove that when

doi:10.1016/j.disc.2007.08.029
fatcat:vovbvqhmrnevvispxul3swbtye
*q*is any odd prime power, the*distance*-2*graph*on the set of vertices at maximal*distance*D from any fixed vertex of the Hemmeter*graph*Hem D (*q*) is isomorphic to the*graph*Quad D−1 ... (*q*) of quadratic forms on F D−1*q*. ...*Bipartite**Q*-*polynomial**distance*-*regular**graphs*In this section, we recall some recent results concerning*bipartite**Q*-*polynomial**distance*-*regular**graphs*. ...##
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Twice Q-polynomial distance-regular graphs of diameter 4

2014
*
Science China Mathematics
*

It is known that a

doi:10.1007/s11425-014-4958-0
fatcat:gqkmpi7wxrhslmeqliq75d6bby
*distance*-*regular**graph*with valency k at least three admits at most two*Q*-*polynomial*structures.*distance*-*regular**graphs*with diameter four and valency at least three admitting two*Q*-*polynomial*... By the work of Dickie Dickie this implies that any*distance*-*regular**graph*with diameter d at least four and valency at least three admitting two*Q*-*polynomial*structures is, provided it is not a Hadamard ...*Distance*-*regular**graphs*of diameter 2 are strongly*regular**graphs*, which possess two P -*polynomial*and two*Q*-*polynomial*structures. ...##
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Distance-regular graphs
[article]

2016
*
arXiv
*
pre-print

This is a survey of

arXiv:1410.6294v2
fatcat:s6kq2ydbtzaldcwjk4qgawfawu
*distance*-*regular**graphs*. ... We present an introduction to*distance*-*regular**graphs*for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of*distance*-*regular**graphs*since the ... Section 5.4) for*bipartite**Q*-*polynomial**distance*-*regular**graphs*. ...##
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On Middle Cube Graphs

2015
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Electronic Journal of Graph Theory and Applications
*

Here we study some of the properties of the middle cube

doi:10.5614/ejgta.2015.3.2.3
fatcat:cyamycmgcvaetjrxzp2mkvx7ve
*graphs*in the light of the theory of*distance*-*regular**graphs*. ... The middle cube*graph*of parameter k is the subgraph of*Q*2k−1 induced by the set of vertices whose binary representation has either k − 1 or k number of ones. ... Compare the Figs. 5 and 7 in order to realize the isomorphism between the definitions of M*Q*3 and O 3 . It is known that O k is a*bipartite*2-antipodal*distance*-*regular**graph*. ...
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