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Dual Bipartite Q-polynomial Distance-regular Graphs
1996
European journal of combinatorics (Print)
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Let ⌫ denote a 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 . ...
doi:10.1006/eujc.1996.0052
fatcat:ydkjuzmyyfcardrctoamjozayi
On bipartite Q-polynomial distance-regular graphs
2007
European journal of combinatorics (Print)
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Let Γ denote a 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] . ...
doi:10.1016/j.ejc.2005.09.003
fatcat:7ge2yrvpwbbnnbzn2sqpcpnvl4
Bipartite Q-polynomial distance-regular graphs and uniform posets
[article]
2011
arXiv
pre-print
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Let 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 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. ...
arXiv:1108.2484v1
fatcat:tfxrnwpntzfqdmze63wbrrsysi
Almost-bipartite distance-regular graphs with the Q-polynomial property
[article]
2005
arXiv
pre-print
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Let G denote a 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. ...
arXiv:math/0508435v1
fatcat:jpu4p4qx6rb6lag2jdgbk3j4xy
Bipartite Q-Polynomial Quotients of Antipodal Distance-Regular Graphs
1999
Journal of combinatorial theory. Series B (Print)
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Let 1 denote a 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. ...
doi:10.1006/jctb.1999.1911
fatcat:f2sbbfywkjgbbhcboj2zy5d7yq
Bipartite Q-polynomial distance-regular graphs and uniform posets
2012
Journal of Algebraic Combinatorics
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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. ...
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. ...
doi:10.1007/s10801-012-0401-1
fatcat:dwea5vh66bgm3n6hw4xb3acwki
Almost-bipartite distance-regular graphs with the Q-polynomial property
2007
European journal of combinatorics (Print)
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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 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. ...
doi:10.1016/j.ejc.2005.07.004
fatcat:mxm27cc6xbg5le53gutydzmkja
On bipartite Q-polynomial distance-regular graphs with c2=1
2007
Discrete Mathematics
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Let denote a 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. ...
doi:10.1016/j.disc.2005.09.044
fatcat:vephvt5isrh4hpt27ngogaor24
The last subconstituent of a bipartite Q-polynomial distance-regular graph
2003
European journal of combinatorics (Print)
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Let Γ denote a 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. ...
doi:10.1016/s0195-6698(03)00059-3
fatcat:kxrdgvurivd3lccs227yzw7iry
On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11
2018
Electronic Journal of Combinatorics
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Let $\Gamma$ denote a 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. ...
doi:10.37236/7347
fatcat:scmlwl37wfbppg7un5qgmzihmi
On bipartite Q-polynomial distance-regular graphs with c 2 2
unpublished
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Let Γ denote a 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. ...
fatcat:5dxqex472bh2vlmcrbyhk2yaji
The last subconstituent of the Hemmeter graph
2008
Discrete Mathematics
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We prove that when 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. ...
doi:10.1016/j.disc.2007.08.029
fatcat:vovbvqhmrnevvispxul3swbtye
Twice Q-polynomial distance-regular graphs of diameter 4
2014
Science China Mathematics
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2014 pre-print arXiv:1405.2546v1 fatcat:dkmkk7wsmzefdjzifcovsgg7zy
It is known that a 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. ...
doi:10.1007/s11425-014-4958-0
fatcat:gqkmpi7wxrhslmeqliq75d6bby
Distance-regular graphs
[article]
2016
arXiv
pre-print
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This is a survey of 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. ...
arXiv:1410.6294v2
fatcat:s6kq2ydbtzaldcwjk4qgawfawu
On Middle Cube Graphs
2015
Electronic Journal of Graph Theory and Applications
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Here we study some of the properties of the middle cube 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. ...
doi:10.5614/ejgta.2015.3.2.3
fatcat:cyamycmgcvaetjrxzp2mkvx7ve
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