Geodesic
Counting $k$-gons in finite projective planes
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 241-270

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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.
Keywords: counting cycles, adjacency matrix, finite projective planes, non-Desarguesian planes. 
@article{SEMR_2013_10_a35,
     author = {A. N. Voropaev},
     title = {Counting $k$-gons in finite projective planes},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {241--270},
     publisher = {mathdoc},
     volume = {10},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2013_10_a35/}
}
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A. N. Voropaev. Counting $k$-gons in finite projective planes. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 10 (2013), pp. 241-270. http://geodesic.mathdoc.fr/item/SEMR_2013_10_a35/