Partial geometry

Let there be an incidence structure $C=(P,L,I)$ ${\ displaystyle C = (P, L, I)}$ ${\ displaystyle C = (P, L, I)}$ consisting of points $P$ ${\ displaystyle P}$ $P$ direct $L$ ${\ displaystyle L}$ $L$ and flags $I\subseteq P\times L$ ${\ displaystyle I \ subseteq P \ times L}$ $I \ subseteq P \ times L$ . They say the point $p$ ${\ displaystyle p}$ $p$ direct incident $l$ ${\ displaystyle l}$ $l$ , if $(p,l)\in I$ ${\ displaystyle (p, l) \ in I}$ $(p, l) \ in I$ . A structure is called finite partial geometry if integers exist $s,t,\alpha \geq 1$ ${\ displaystyle s, t, \ alpha \ geq 1}$ ${\ displaystyle s, t, \ alpha \ geq 1}$ such that:

For any pair of different points $p$ ${\ displaystyle p}$ $p$ and $q$ ${\ displaystyle q}$ $q$ there is a maximum of one line incident to both points.
Every direct incident $s+1$ ${\ displaystyle s + 1}$ $s + 1$ points.
Every point is incidental $t+1$ ${\ displaystyle t + 1}$ $t + 1$ direct.
If the point $p$ ${\ displaystyle p}$ $p$ and direct $l$ ${\ displaystyle l}$ $l$ not incidental, exists exactly $\alpha$ ${\ displaystyle \ alpha}$ $\ alpha$ steam $(q,m)\in I$ ${\ displaystyle (q, m) \ in I}$ ${\ displaystyle (q, m) \ in I}$ such that $p$ ${\ displaystyle p}$ $p$ incidental $m$ ${\ displaystyle m}$ $m$ , but $q$ ${\ displaystyle q}$ $q$ incidental $l$ ${\ displaystyle l}$ $l$ .

Partial geometry with these parameters is denoted by $pg(s,t,\alpha )$ ${\ displaystyle pg (s, t, \ alpha)}$ ${\ displaystyle pg (s, t, \ alpha)}$ .

Content

1 Properties
2 Special cases
3 Generalizations
4 notes
5 Literature

Properties

The number of points is given by the formula ${\frac {(s+1)(st+\alpha )}{\alpha }}$ ${\ displaystyle {\ frac {(s + 1) (st + \ alpha)} {\ alpha}}}$ , and the number of lines by the formula ${\frac {(t+1)(st+\alpha )}{\alpha }}$ ${\ displaystyle {\ frac {(t + 1) (st + \ alpha)} {\ alpha}}}$ .
Dot graph ^[1] structure $pg(s,t,\alpha )$ ${\ displaystyle pg (s, t, \ alpha)}$ is a strongly regular graph : $srg((s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1))$ ${\ displaystyle srg ((s + 1) {\ frac {(st + \ alpha)} {\ alpha}}, s (t + 1), s-1 + t (\ alpha -1), \ alpha (t + one))}$ .
Partial geometries are dual - a dual structure for $pg(s,t,\alpha )$ ${\ displaystyle pg (s, t, \ alpha)}$ is just a structure $pg(t,s,\alpha )$ ${\ displaystyle pg (t, s, \ alpha)}$ .

Special cases

Generalized quadrangles are exactly partial geometries $pg(s,t,\alpha )$ ${\ displaystyle pg (s, t, \ alpha)}$ from $\alpha =1$ ${\ displaystyle \ alpha = 1}$ .
Steiner systems are exactly partial geometries $pg(s,t,\alpha )$ ${\ displaystyle pg (s, t, \ alpha)}$ from $\alpha =s+1$ ${\ displaystyle \ alpha = s + 1}$ .

Generalizations

$S=(P,L,I)$ ${\ displaystyle S = (P, L, I)}$ of order $s, t$ ${\ displaystyle s, t}$ called semi-partial geometry if integers exist $\alpha \geq 1,\mu$ ${\ displaystyle \ alpha \ geq 1, \ mu}$ such that:

If the point $p$ ${\ displaystyle p}$ and direct $\ell$ ${\ displaystyle \ ell}$ not incidental, exists either $0$ ${\ displaystyle 0}$ or exactly $\alpha$ ${\ displaystyle \ alpha}$ steam $(q,m)\in I$ ${\ displaystyle (q, m) \ in I}$ such that $p$ ${\ displaystyle p}$ incidental $m$ ${\ displaystyle m}$ and $q$ ${\ displaystyle q}$ incidental $\ell$ ${\ displaystyle \ ell}$ .
Any pair of noncollinear points has exactly $\mu$ ${\ displaystyle \ mu}$ common neighbors.

A semi-partial geometry is partial geometry if and only if $\mu =\alpha (t+1)$ ${\ displaystyle \ mu = \ alpha (t + 1)}$ .

It is easy to show that the collinearity graph ^{[1] of} such a geometry is strictly regular with parameters $(1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1),s-1+t(\alpha -1),\mu )$ ${\ displaystyle (1 + s (t + 1) + s (t + 1) t (s- \ alpha +1) / \ mu, s (t + 1), s-1 + t (\ alpha -1) , \ mu)}$ .

A good example of this geometry is obtained by taking affine points $PG(3,q^{2})$ ${\ displaystyle PG (3, q ^ {2})}$ and only those lines that intersect the plane at infinity at a point of a fixed Baire subplane. Geometry has parameters $(s,t,\alpha ,\mu )=(q^{2}-1,q^{2}+q,q,q(q+1))$ ${\ displaystyle (s, t, \ alpha, \ mu) = (q ^ {2} -1, q ^ {2} + q, q, q (q + 1))}$ .

Notes

↑ ¹ ² If a partial geometry P is given in which any two points define at most one straight line, a collinearity graph or a point graph of geometry P is a graph whose vertices are points P and two vertices are connected by an edge if and only if they define a straight line in P.

Literature

Brouwer AE, van Lint JH Strongly regular graphs and partial geometries // Enumeration and Design / Jackson DM, Vanstone SA. - Toronto: Academic Press, 1984. - S. 85–122.
Bose RC Strongly regular graphs, partial geometries and partially balanced designs // Pacific J. Math. - 1963.- T. 13 . - S. 389-419 .
De Clerck F., Van Maldeghem H. Some classes of rank 2 geometries // Handbook of Incidence Geometry. - Amsterdam: North-Holland, 1995. - S. 433-475.
Thas JA Partial Geometries // Handbook of Combinatorial Designs / Colbourn Charles J., Dinitz Jeffrey H .. - 2nd. - Boca Raton: Chapman & Hall / CRC, 2007. - S. 557-561. - ISBN 1-58488-506-8 .
Debroey I., Thas JA On semipartial geometries // Journal of Combinatorial Theory Ser. A. - 1978. - T. 25 . - S. 242–250 .

[point_graph-1] ¹ ² If a partial geometry P is given in which any two points define at most one straight line, a collinearity graph or a point graph of geometry P is a graph whose vertices are points P and two vertices are connected by an edge if and only if they define a straight line in P.