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Partial geometry
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<h2 id="properties">Properties</h2> <ul><li>The number of points is given by ( s + 1 ) ( s t + α ) α {\displaystyle {\frac {(s+1)(st+\alpha )}{\alpha }}} and the number of lines by ( t + 1 ) ( s t + α ) α {\displaystyle {\frac {(t+1)(st+\alpha )}{\alpha }}} .</li> <li>The point graph (also known as the <a href="/facts/Collinearity/SxbVBJYR">collinearity graph</a>) of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is a <a href="/facts/Strongly_regular_graph/0GbdLwSd">strongly regular graph</a>: s r g ( ( s + 1 ) ( s t + α ) α , s ( t + 1 ) , s − 1 + t ( α − 1 ) , α ( t + 1 ) ) {\displaystyle \mathrm {srg} {\Big (}(s+1){\frac {(st+\alpha )}{\alpha }},s(t+1),s-1+t(\alpha -1),\alpha (t+1){\Big )}} .</li> <li>Partial geometries are dualizable structures: the dual of a p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} is simply a p g ( t , s , α ) {\displaystyle \mathrm {pg} (t,s,\alpha )} .</li></ul> <h2 id="special-cases">Special cases</h2> <ul><li>The <a href="/facts/Generalized_quadrangle/nFRoYCIc">generalized quadrangles</a> are exactly those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with α = 1 {\displaystyle \alpha =1} .</li> <li>The <a href="/facts/Steiner_system/DdakpJpm">Steiner systems</a> S ( 2 , s + 1 , t s + 1 ) {\displaystyle S(2,s+1,ts+1)} are precisely those partial geometries p g ( s , t , α ) {\displaystyle \mathrm {pg} (s,t,\alpha )} with α = s + 1 {\displaystyle \alpha =s+1} .</li></ul> <h2 id="generalisations">Generalisations</h2> <p>A <a href="/facts/Partial_linear_space/F5PXLKDB">partial linear space</a> S = ( P , L , I ) {\displaystyle S=(P,L,I)} of order s , t {\displaystyle s,t} is called a semipartial geometry if there are <a href="/facts/Integer/PUwwrawx">integers</a> α ≥ 1 , μ {\displaystyle \alpha \geq 1,\mu } such that: </p> <ul><li>If a point p {\displaystyle p} and a line l {\displaystyle l} are not incident, there are either 0 {\displaystyle 0} or exactly α {\displaystyle \alpha } pairs ( q , m ) ∈ I {\displaystyle (q,m)\in I} , such that p {\displaystyle p} is incident with m {\displaystyle m} and q {\displaystyle q} is incident with l {\displaystyle l} .</li> <li>Every pair of non-collinear points have exactly μ {\displaystyle \mu } common neighbours.</li></ul> <p>A semipartial geometry is a partial geometry if and only if μ = α ( t + 1 ) {\displaystyle \mu =\alpha (t+1)} . </p><p>It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters ( 1 + s ( t + 1 ) + s ( t + 1 ) t ( s − α + 1 ) / μ , s ( t + 1 ) , s − 1 + t ( α − 1 ) , μ ) {\displaystyle (1+s(t+1)+s(t+1)t(s-\alpha +1)/\mu ,s(t+1),s-1+t(\alpha -1),\mu )} . </p><p>A nice example of such a geometry is obtained by taking the affine points of P G ( 3 , q 2 ) {\displaystyle \mathrm {PG} (3,q^{2})} and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters ( s , t , α , μ ) = ( 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))} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Strongly_regular_graph/0GbdLwSd">Strongly regular graph</a></li> <li><a href="/facts/Maximal_arc/WByc4hPJ">Maximal arc</a></li></ul> <ul><li>Brouwer, A.E.; van Lint, J.H. (1984), "Strongly regular graphs and partial geometries", in Jackson, D.M.; Vanstone, S.A. (eds.), <i>Enumeration and Design</i>, Toronto: Academic Press, pp. 85–122</li> <li>Bose, R. C. (1963), <a href="https://repository.lib.ncsu.edu/bitstream/1840.4/2478/1/ISMS_1963_358.pdf">"Strongly regular graphs, partial geometries and partially balanced designs"</a> (PDF), <i>Pacific J. Math.</i>, 13: 389–419, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1963.13.389">10.2140/pjm.1963.13.389</a></li> <li>De Clerck, F.; Van Maldeghem, H. (1995), "Some classes of rank 2 geometries", <i>Handbook of Incidence Geometry</i>, Amsterdam: North-Holland, pp. 433–475</li> <li>Thas, J.A. (2007), "Partial Geometries", in Colbourn, Charles J.; Dinitz, Jeffrey H. (eds.), <a href="https://archive.org/details/handbookofcombin0000unse/page/557"><i>Handbook of Combinatorial Designs</i></a> (2nd ed.), Boca Raton: Chapman & Hall/ CRC, pp. <a href="https://archive.org/details/handbookofcombin0000unse/page/557">557–561</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-58488-506-8</li> <li>Debroey, I.; Thas, J. A. (1978), "On semipartial geometries", <i>Journal of Combinatorial Theory, Series A</i>, 25: 242–250, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0097-3165%2878%2990016-x">10.1016/0097-3165(78)90016-x</a></li></ul>