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. 
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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/

[1] D. R. Hughes, F. C. Piper, Projective Planes, Springer, Berlin, 1973 | MR

[2] M. Hall, Jr., J. D. Swift, R. J. Walker, “Uniqueness of the Projective Plane of Order Eight”, Mathematical Tables and Other Aids to Computation, 10 (1956), 186–194 | DOI | MR | Zbl

[3] C. W. H. Lam, L. Thiel, S. Swiercz, “The Non-Existence of Finite Projective Planes of Order 10”, Canadian Journal of Mathematics, XLI (1989), 1117–1123 | DOI | MR | Zbl

[4] C. W. H. Lam, G. Kolesova, L. Thiel, “A Computer Search for Finite Projective Planes of Order 9”, Discrete Mathematics, 92 (1991), 187–195 | DOI | MR | Zbl

[5] Projective Planes of Small Order, http://www.uwyo.edu/moorhouse/pub/planes

[6] F. Lazebnik, K. E. Mellinger, O. Vega, “On the Number of $k$-gons in Finite Projective Planes”, Note di Matematica, 29 (2009), 135–152 | MR

[7] A. N. Voropaev, “Vyvod yavnykh formul dlya podschëta tsiklov fiksirovannoi dliny v neorientirovannykh grafakh”, Informatsionnye protsessy, 11 (2011), 90–113

[8] A. N. Voropaev, “Podschët tsiklov v dvudolnykh grafakh s dlinoi menee trëkh obkhvatov”, Informatsionnye protsessy, 11 (2011), 500–509

[9] F. Kharari, Teoriya grafov, Mir, M., 1973 | MR

[10] A. N. Voropaev, “Uchët obkhvata pri podschëte korotkikh tsiklov v dvudolnykh grafakh”, Informatsionnye protsessy, 11 (2011), 225–252

[11] A. N. Voropaev, “Kratnosti summ v yavnykh formulakh dlya podschëta tsiklov fiksirovannoi dliny v neorientirovannykh grafakh”, Prikladnaya diskretnaya matematika, 14 (2011), 42–55

[12] N. Alon, R. Yuster, U. Zwick, “Finding and Counting Given Length Cycles”, Algorithmica, 17 (1997), 209–223 | DOI | MR | Zbl

[13] Explicit formulae: FlowProblem, http://flowproblem.ru/cycles/explicit-formulae

[14] A. M. Karavaev, A. N. Voropaev, “Effektivnost rasparallelivaniya yavnykh formul dlya podschëta korotkikh tsiklov v grafe”, Parallelnye vychislitelnye tekhnologii, Mezhdunarodnaya nauchnaya konferentsiya PaVT'2010, Izd. tsentr YuUrGU, Chelyabinsk, 2010, 486–497

[15] T. Penttila, G. F. Royle, M. K. Simpson, “Hyperovals in the Known Projective Planes of Order 16”, Journal of Combinatorial Designs, 4 (1996), 59–65 | 3.0.CO;2-Z class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[16] H. Neumann, “On Some Finite Non-Desarguesian Planes”, Archiv der Mathematik, 6 (1954), 36–40 | DOI | MR | Zbl