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### Dual Bipartite Q-polynomial Distance-regular Graphs

Garth A. Dickie, Paul M. Terwilliger
1996 European journal of combinatorics (Print)
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 .  ...

### On bipartite Q-polynomial distance-regular graphs

Štefko Miklavič
2007 European journal of combinatorics (Print)
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] .  ...

### Bipartite Q-polynomial distance-regular graphs and uniform posets [article]

Stefko Miklavic, Paul Terwilliger
2011 arXiv   pre-print
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.  ...

### Almost-bipartite distance-regular graphs with the Q-polynomial property [article]

Michael S. Lang, Paul M. Terwilliger
2005 arXiv   pre-print
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.  ...

### Bipartite Q-Polynomial Quotients of Antipodal Distance-Regular Graphs

John S. Caughman
1999 Journal of combinatorial theory. Series B (Print)
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.  ...

### Bipartite Q-polynomial distance-regular graphs and uniform posets

Štefko Miklavič, Paul Terwilliger
2012 Journal of Algebraic Combinatorics
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.  ...

### Almost-bipartite distance-regular graphs with the Q-polynomial property

Michael S. Lang, Paul M. Terwilliger
2007 European journal of combinatorics (Print)
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.  ...

### On bipartite Q-polynomial distance-regular graphs with c2=1

Štefko Miklavič
2007 Discrete Mathematics
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.  ...

### The last subconstituent of a bipartite Q-polynomial distance-regular graph

John S. Caughman
2003 European journal of combinatorics (Print)
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.  ...

### On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

Štefko Miklavič
2018 Electronic Journal of Combinatorics
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.  ...

### On bipartite Q-polynomial distance-regular graphs with c 2 2

Stefko Miklavič, Safet Penji´c
unpublished
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.  ...

### The last subconstituent of the Hemmeter graph

John S. Caughman, IV, Elizabeth J. Hart, Jianmin Ma
2008 Discrete Mathematics
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.  ...

### Twice Q-polynomial distance-regular graphs of diameter 4

JianMin Ma, Jack H. Koolen
2014 Science China Mathematics
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.  ...

### Distance-regular graphs [article]

Edwin R. van Dam, Jack H. Koolen, Hajime Tanaka
2016 arXiv   pre-print
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.  ...

### On Middle Cube Graphs

Cristina Dalfo, Miquel Angel Fiola, Margarida Mitjana
2015 Electronic Journal of Graph Theory and Applications
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.  ...
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