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Generalized quadrangles and regularity

Matthew R. Brown
2005 Discrete Mathematics  
Let S = (P, B, I) be a generalized quadrangle of order (s, t), s, t > 1, and assume that S has a regular point X.  ...  In this paper we survey some basic results on such generalized quadrangles (GQs) as well as the known examples.  ...  In this case N X is called a skew translation generalized quadrangle (see [23] ).  ... 
doi:10.1016/j.disc.2004.04.034 fatcat:t6njkpxo2fdkfasifwaxradj6a

Semi-regular graph automorphisms and generalized quadrangles

Mark A Ronan
1980 Journal of combinatorial theory. Series A  
AN APPLICATION TO GENERALIZED QUADRANGLES Introduction and Theorems Recall that a generalized quadrangle Q is a geometry of points and lines such that two points lie on at most one line, and given a  ...  Root Groups and Moufalzg Quadrangles If Q is a generalized quadrangle, a root p is a chain of length 4 in the flaggraph of Q.  ... 
doi:10.1016/0097-3165(80)90026-6 fatcat:i7jmchjabjh2zmhdqa6k66q4cy

Generalized quadrangles with a regular point and association schemes

Dina Ghinelli, Stefan Löwe
1995 Linear Algebra and its Applications  
Here we first construct from a generalized quadrangle d with a regular point a four-class association scheme .&@'I called in brief geometric.  ...  There is a new method of constructing generalized quadrangles (GQs) which is based on covering of nets; all GQs with a regular point can be represented in this way.  ...  THE GEOMETRIC ASSOCIATION SCHEME Let d be a generalized quadrangle of order (s, t) with s > t > 2 and regular point 00.  ... 
doi:10.1016/0024-3795(94)00305-w fatcat:4qc2ysz6ofg2lepqobyv2o67pi

Ovoids of Generalized Quadrangles of Order (q, q^2-q) and Delsarte Cocliques in Related Strongly Regular Graphs [article]

Mohammad Adm, Ryan Bergen, Ferdinand Ihringer, Sam Jaques, Karen Meagher, Alison Purdy, Boting Yang
2021 arXiv   pre-print
We show that Delsarte cocliques do not exist for all Taylor's 2-graphs and for point graphs of generalized quadrangles of order (q,q^2-q) for infinitely many q.  ...  We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal.  ...  Haemers, Koen Thas and Joseph A. Thas for pointing us to various references, in particular [10] and [13] . We would like to thank Alexander L. Gavrilyuk for pointing out the idea for Lemma 7.1.  ... 
arXiv:1705.02066v4 fatcat:wsypfw33ira2bluyte2djitoli

New Combinatorial Characterizations of Generalized Quadrangles

J.A. Thas
1981 European journal of combinatorics (Print)  
vi) and if s ~ 1 ~t, it is called a Moufang generalized quadrangle [12] , and by a celebrated theorem of Tits [11, 13] a thick generalized quadrangle is a Moufang generalized quadrangle iff it is one  ...  Since s is odd, 0 is an elliptic quadric [1], and so T(O) =0(5, s). Hence Y=H(3, s).(b) Let Y be a thick generalized quadrangle with s ~ t and all points regular.  ... 
doi:10.1016/s0195-6698(81)80035-2 fatcat:zir7y4tvlze33dks2ob2nlbemy

Regularizing quadrangles in the Möbius plane

Georg Eberharter, Johann Lang, Otto Röschel
2018 Journal of Geometry  
The regularizing effect also depends upon the shape of the starting quadrangle. The investigation of such shapes leads into the field of complex dynamics in one variable.  ...  For any given cross ratio δ ∈ C we define a non-linear Möbiusinvariant procedure creating vertices of a new quadrangle from a given quadrangle.  ...  The authors thank the editor and the reviewers for valuable suggestions for improvement.  ... 
doi:10.1007/s00022-018-0453-z fatcat:rrrgqo7fpnh27go2oageodx7va

MOUFANG QUADRANGLES OF MIXED TYPE

KOEN STRUYVE, HENDRIK VAN MALDEGHEM
2008 Glasgow Mathematical Journal  
Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young  ...  In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular.  ...  It is easy to see that is a generalized quadrangle. Also, since is regular, is also regular.  ... 
doi:10.1017/s0017089507004016 fatcat:bigpdoosu5ae5lqumnzjmm7izu

Moufang quadrangles of mixed type [article]

Koen Struyve, Hendrik Van Maldeghem
2009 arXiv   pre-print
Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual net associated to the regular points satisfy the Axiom of Veblen-Young  ...  In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular.  ...  Corollary 3.1 (i) A generalized quadrangle Γ is a mixed quadrangle or a symplectic quadrangle in characteristic 2 if and only if all its points are regular, at least one line is regular, and Γ is isomorphic  ... 
arXiv:0909.3208v1 fatcat:nis5evqxt5g5ddkb7r772jatw4

Page 5492 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
,q), ¢ #1 and q odd, are characterized in terms of 3-regularity, and a pure geometrical construction of flock generalized quadrangles is given.  ...  Here we study the impli- cations of having 3-regular triples; in particular, we characterize classes of generalized quadrangles in terms of 3-regularity.  ... 

Generalized quadrangles as amalgamations of projective planes

Stanley E Payne
1972 Journal of Algebra  
The known examples of generalized quadrangles based on finite fields of even characteristic have all one notable feature in common-a regular point (co) incident with a regular line [co] (cf., Section 3  ...  Let 171 and 17, be two projective planes with exactly the same set of points.  ...  REGULAR POINTS AND LINES Let P be a generalized quadrangle of order s. Let X and Y be distinct points of P, and let 2, , 2, ,..., Z, be the points of P collinear with both X and Y.  ... 
doi:10.1016/0021-8693(72)90108-1 fatcat:acyqh42xtvfdngcmmijlobe5lm

Pseudo-Geometric Strongly Regular Graphs with a Regular Point [article]

Edwin van Dam, Krystal Guo
2022 arXiv   pre-print
We give an explicit construction for q new, pairwise non-isomorphic graphs with the same parameters as the collinearity graph of generalized quadrangles of order (q,q) and a new non-geometric graph with  ...  the same parameters as the collinearity graph of the Hermitian generalized quadrangle of order (q^2, q), for prime powers q.  ...  A cospectral mate for the Hermitian generalized quadrangle For s > t ≥ 2, the known generalized quadrangles of order (s, t) with a regular point all have s = q 2 and t = q, for prime powers q.  ... 
arXiv:2204.04755v1 fatcat:67j2tkjdwffbtn575rchprbndu

Quantum Walks on Generalized Quadrangles [article]

Chris Godsil, Krystal Guo, Tor G.J. Myklebust
2015 arXiv   pre-print
We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of order (5^2,5) under this matrix and thus provide strongly regular counter-examples  ...  It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of S^+(U^3), a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs.  ...  Generalized quadrangles The spectrum of S + (U 3 ) distinguishes strongly regular graphs for many parameter sets.  ... 
arXiv:1511.01962v1 fatcat:yyiwglakvfbijgcfidlyluzxwu

All generalized quadrangles of order 3 are known

Stanley E Payne
1975 Journal of combinatorial theory. Series A  
A generalized quadrangle of order 3 must be isomorphic either to the quadrangle p4 or to its dual, where B, consists of all points of PG(3,3) and those lines of PG(3,3) self-conjugate with respect to a  ...  Throughout this note 8, will denote a generalized quadrangle of orders, i.e., a partial plane with 1 + s points (lines) incident with each line (point) and satisfying the following axiom: If x is a point  ...  It is well known and easy to show that any quadrangle of order 2 has all pairs of points (lines) regular, so is unique.  ... 
doi:10.1016/0097-3165(75)90009-6 fatcat:5mjtqhqponhsjdbaqblbju33ju

Page 1033 of Mathematical Reviews Vol. 54, Issue 4 [page]

1977 Mathematical Reviews  
The notion of a “regular [antiregular] pair of points” is explained and a characterization of such pairs in generalized quadrangles of order s=t>1 (due to Payne) is cited.  ...  In Section III collineations of generalized quadrangles of order (s, t) are studied and restrictions for the parameters of self-dual generalized quadrangles are given.  ... 

Finite Moufang generalized quadrangles

J. A. Thas, H. Van Maldeghem
2006 Journal of Mathematical Sciences  
Theorem 4. 6 6 A finite generalized quadrangle of order s, with s = 1, is isomorphic to W (s) if and only if all its points are regular.  ...  Theorem 6. 4 4 Let S be a translation generalized quadrangle of order (s, s 2 ), s = 1, with translation point (∞) and having a regular line M not incident with the point (∞).  ... 
doi:10.1007/s10958-006-0287-3 fatcat:cqio3qr5iffq3pmacnbzvrsnwa
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