Combinatorics of Partial Geometries and Generalized Quadrangles

A finite partial geometry [17] is an incidence structure S=(P,B,I), with a symmetric incidence relation satisfying the following axioms (i) each point is incident with 1+t lines (t⩾1)and two distinct points are incident with at most one line; (ii) each line is incident with 1+s points (s⩾1) and two distinct lines are incident with at most one point; (iii) if x is a point and L is a line not incident with x, then there are exactly α (α ⩾1) points x1 ,x2 ,…,xα and α lines L1 ,L2 ,…, Lα such that xILi Ixi IL, i=1,2,…,α.

[1]  B. Segre.  Ovals In a Finite Projective Plane , 1955, Canadian Journal of Mathematics.

[2]  Jacques Tits,et al.  Sur la trialité et certains groupes qui s’en déduisent , 1959 .

[3]  Beniamino Segre,et al.  Lectures on modern geometry , 1962 .

[4]  J. Tits.  Ovoßdes et Groupes de Suzuki , 1962 .

[5]  R. C. Bose.  Strongly regular graphs, partial geometries and partially balanced designs. , 1963 .

[6]  Walter Feit,et al.  The nonexistence of certain generalized polygons , 1964 .

[7]  J. Seidel.  Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3 , 1968 .

[8]  R. Denniston.  Some maximal arcs in finite projective planes , 1969 .

[9]  Francis Buekenhout.  Une caractérisation des espaces affins basée sur la notion de droite , 1969 .

[10]  George Szekeres,et al.  On a combinatorial generalization of 27 lines associated with a cubic surface , 1969, Journal of the Australian Mathematical Society.

[11]  C. T. Benson,et al.  On the structure of generalized quadrangles , 1970 .

[12]  Stanley E. Payne.  The equivalence of certain generalized quadrangles , 1971 .

[13]  Stanley E. Payne,et al.  Nonisomorphic generalized quadrangles , 1971 .

[14]  E. Shult.  Characterizations of certain classes of graphs , 1972 .

[15]  R. C. Bose,et al.  Geometric and pseudo-geometric graphs (q2+1,q+1,1) , 1972 .

[16]  Joseph A. Thas,et al.  Ovoidal translation planes , 1972 .

[17]  Stanley E. Payne,et al.  Quadrangles of order (s − 1, s + 1) , 1972 .

[18]  A restriction on the parameters of a subquadrangle , 1973 .

[19]  Walter D. Wallis,et al.  Configurations Arising from Maximal Arcs , 1973, J. Comb. Theory, Ser. A.

[20]  Stanley E. Payne.  Generalized Quadrangles of Even Order , 1974 .

[21]  Claude Lefèvre,et al.  Generalized quadrangles in projective spaces , 1974 .

[22]  On polarities of symmetric partial geometries , 1974 .

[23]  J. Thas.  4-Gonal configurations with parameters r=q2+1 and k=q+1 , 1974 .

[24]  J. Thas.  Construction of maximal arcs and partial geometries , 1974 .

[25]  Stanley E. Payne,et al.  All Generalized Quadrangles of Order 3 Are Known , 1975, J. Comb. Theory A.

[26]  Joseph A. Thas.  Some Results Concerning {(q + 1)(n - 1); n}-Arcs and {(q + 1)(n - 1) + 1; n}-Arcs in Fintie Projective Planes of Order q , 1975, J. Comb. Theory, Ser. A.

[27]  J. Thas.  On generalized quadrangles with parameters s=q2 and t=q3 , 1976 .

[28]  Generalized quadrangles in finite projective spaces , 1977 .

[29]  J. Thas.  Combinatorial characterizations of the classical generalized quadrangles , 1977 .

[30]  J. Thas.  Combinatorial characterizations of generalized quadrangles with parameters s = q and t = q2 , 1978 .