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A classification of one-dimensional affine rank three graphs. (English) Zbl 1466.05227

Summary: The rank three subgroups of a one-dimensional affine group over a finite field were classified in [D. A. Foulser and M. J. Kallaher, Geom. Dedicata 7, 111–130 (1978; Zbl 0406.51009)]. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by J. H. van Lint and A. Schrijver [Combinatorica 1, 63–73 (1981; Zbl 0491.05018)] or by W. Peisert [J. Algebra 240, No. 1, 209–229 (2001; Zbl 1021.05051)]. Our approach is based on elementary group theory and does not use the classification of rank three affine groups.

MSC:

05E30 Association schemes, strongly regular graphs
51E15 Finite affine and projective planes (geometric aspects)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:

[1] Andries E. Brouwer, Hendrik van Maldeghem, Strongly regular graphs, https://homepages.cwi.nl/ aeb/math/srg/rk3/srgw.pdf.
[2] Foulser, D. A.; Kallaher, M. J., Solvable, flag-transitive, rank 3 collineation groups, Geom. Dedicata, 7, 111-130 (1978) · Zbl 0406.51009
[3] Jones, Gareth, Paley and the Paley graphs, (Isomorphisms, Symmetry and Computations in Algebraic Graph Theory. Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, Springer Proceedings in Mathematics & Statistics, vol. 305 (2020)), 155-184 · Zbl 1442.05229
[4] Lim, Tian Khoon; Praeger, Cheryl E., On generalised Paley graphs and their automorphism groups, Michigan Math. J., 58, 1, 293-308 (2009) · Zbl 1284.05175
[5] Muzychuk, Mikhail; Ponomarenko, Ilia, Schur rings, European J. Combin., 30, 1526-1539 (2009) · Zbl 1195.20003
[6] Peisert, W., All self-complementary symmetric graphs, J. Algebra, 240, 209-229 (2001) · Zbl 1021.05051
[7] van Lint, J. H.; Schrijver, A., Constructions of strongly regular graphs, two-weight codes and partial geometries by finite fields, Combinatorica, 1, 63-73 (1981) · Zbl 0491.05018
[8] Wielandt, H., Finite Permutation Groups (1964), Academic Press: Academic Press New York · Zbl 0138.02501
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