A note on almost partial geometries (Q1268373)

An almost partial geometry (APG) is a connected incidence structure with \(v\) points and \(b\) lines that satisfy: (1) there are \(k>1\) points on each line; (2) each point is on \(r>1\) lines (two points are on at most 1 line); (3) for each point \(P\) there exists a unique line \(P^\alpha\) with \(P\notin P^\alpha\) such that the number of points on \(P^\alpha\) joint to \(P\) is a constant \(e\) and for any other line \(l\) with \(P\notin l\neq P^\alpha\) the number of points on \(l\) joint to \(P\) is a constant \(t>0\). The APG is called symmetric if \(k=r\). If \(e=t\) then an APG is a partial geometry. We shall call an APG proper if \(e\neq t\). In this paper the following result is proved: A proper APG is symmetric with \(e\) even.