Geodesic
On the automorphism group of the Aschbacher graph
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 162-176
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A Moore graph is a regular graph of degree $k$ and diameter $d$ with $v$ vertices such that $v\le1+k+k(k-1)+\dots+k(k-1)^{d-1}$. It is known that a Moore graph of degree $k\ge3$ has diameter 2, i.e., it is strongly regular with parameters $\lambda=0$, $\mu=1$ and $v=k^2+1$, where the degree $k$ is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree $k=57$. Aschbacher showed that a Moore graph with $k=57$ is not a graph of rank 3. In this connection, we call a Moore graph with $k=57$ the Aschbacher graph and investigate its automorphism group $G$ without additional assumptions (earlier, it was assumed that $G$ contains an involution).
Mots-clés : automorphism group of a graph
Keywords: Moore graph, strongly regular graph. 
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A. A. Makhnev; D. V. Paduchikh. On the automorphism group of the Aschbacher graph. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 162-176. http://geodesic.mathdoc.fr/item/TIMM_2009_15_2_a14/

[1] Damerell R. M., “On Moore graphs”, Math. Proc. Cambr. Phil. Soc., 74 (1973), 227–236 | DOI | MR | Zbl

[2] Aschbacher M., “The nonexistence of rank three permutation groups of degree 3250 and subdegree 57”, J. Algebra, 19:3 (1971), 538–540 | DOI | MR | Zbl

[3] Makhnev A. A., Paduchikh D. V., “Ob avtomorfizmakh grafa Ashbakhera”, Algebra i logika, 40:2 (2001), 125–134 | MR | Zbl

[4] Brouwer A. E., Haemers W. H., “The Gewirtz graph: an exercise in the theory of graph spectra”, Europ. J. Comb., 14:5 (1993), 397–407 | DOI | MR | Zbl

[5] Cameron P., Permutation groups, London Math. Soc. Stud. Texts, 45, Cambr. Univ. Press, Cambridge, 1999, 220 pp. | Zbl

[6] Cameron P., Van Lint J., Designs, graphes, codes and their links, London Math. Soc. Stud. Texts, 22, Cambr. Univ. Press, Cambridge, 1989, 240 pp. | MR