Geodesic
On the generating graph of direct powers of a simple group
Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 329-350.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let S be a nonabelian finite simple group and let n be an integer such that the direct product S $^{ n }$ is 2-generated. Let $\Gamma (S ^{ n })$ be the generating graph of S $^{ n }$ and let $\Gamma _{ n }(S)$ be the graph obtained from $\Gamma (S ^{ n })$ by removing all isolated vertices. A recent result of Crestani and Lucchini states that $\Gamma _{ n }(S)$ is connected, and in this note we investigate its diameter. A deep theorem of Breuer, Guralnick and Kantor implies that $diam(\Gamma _{1}(S))=2$, and we define $\Delta (S)$ to be the maximal n such that $diam(\Gamma _{ n }(S))=2$. We prove that $\Delta (S)\geq 2$ for all S, which is best possible since $\Delta (A _{5})$=2, and we show that $\Delta (S)$ tends to infinity as |S| tends to infinity. Explicit upper and lower bounds are established for direct powers of alternating groups.
Keywords: finite simple groups, generating graph, diameter, spread 
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     title = {On the generating graph of direct powers of a simple group},
     journal = {Journal of Algebraic Combinatorics},
     pages = {329--350},
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     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2013__38_2_a7/}
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Burness, Timothy C.; Crestani, Eleonora. On the generating graph of direct powers of a simple group. Journal of Algebraic Combinatorics, Tome 38 (2013) no. 2, pp. 329-350. http://geodesic.mathdoc.fr/item/JAC_2013__38_2_a7/