ON PSEUDOGEOMETRIC GRAPHS-OF THE PARTIAL GEOMETRIES PG(2)(4,T)

An incidence system consisting of points and lines is called an alpha- partial geometry of order (s, t) if each line contains s + 1 points, e ach point lies on t + 1 lines (the lines intersect in at most one poin t), and for any point a not lying on a line L there are exactly cu lin es passing through a and intersecting L (this geometry is denoted by p G alpha(s, t)). The point graph of the partial geometry p G(alpha)(s, t) is strongly regular with parameters v = (s + 1)(1 + st/alpha), k = s(t + 1), lambda = (s - 1) + (alpha - 1)t, and mu = alpha(t + 1). A g raph with the indicated parameters is called a pseudogeometric graph o f the corresponding geometry. It is proved that a pseudogeometric grap h of a partial geometry pG(2)(4, t) in which the mu-subgraphs are regu lar graphs without triangles is the triangular graph T(5), the quotien t of the Johnson graph J(8,4), or the McLaughlin graph.