On the number of $k$-gons in finite projective planes

Abstract

Let $\pi = \pi _q$ denote a finite projective plane of order $q$,  and let $G = Levi (\pi)$ be the bipartite point-line incidence graph of  $\pi$.  For $k\ge 3$,  let $c_{2k} (\pi)$ denote the number of  cycles of length $2k$ in $G$.   Are the numbers $c_{2k} (\pi)$ the same for all $\pi _q$?  We prove that this is the case for $k=3,4,5,6$ by computing these numbers.