Counting $k$-gons in finite projective planes

In the study of combinatorial properties of finite projective planes, an open problem is to determine whether the number of $k$-gons in a plane depends on its structure. For the values of $k = 3, 4, 5, 6$, the number of $k$-gons is a function of plane's order $q$ only. By means of the explicit formulae for counting $2\,k$-cycles in bipartite graphs of girth at least 6 derived in this work for the case $k \leqslant 10$, we computed the numbers of $k$-gons in the form of polynomials in plane's order up to the value of $k = 10$. Some asymptotical properties of the numbers of $k$-gons when $q \to \infty$ were also discovered. Our conjectured value of $k$ such that the numbers of $k$-gons in non-isomorphic planes of the same order may differ is 14.