Geodesic
The non-isolated vertices in the generating graph of a direct powers of simple groups.
Journal of Algebraic Combinatorics, Tome 37 (2013) no. 2, pp. 249-263.

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

Summary: For a finite group G let $\Gamma (G)$ denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Many deep results on the generation of the finite simple groups G can be equivalently stated as theorems that ensure that $\Gamma (G)$ is a rich graph, with several good properties. In this paper we want to consider $\Gamma (G ^{ \delta })$ where G is a finite non-abelian simple group and G $^{ \delta }$ is the largest 2-generated power of G, with the aim to investigate whether the good generation properties of G still affect the behaviour of $\Gamma (G ^{ \delta })$. In particular we prove that the graph obtained from $\Gamma (G ^{ \delta })$ by removing the isolated vertices is 1-arc transitive and connected and we investigate the diameter of this graph. Moreover, some intriguing open questions will be introduced and their solutions will be exemplified for $G=\operatorname{Alt}(5)$ .
Keywords: generating graph, direct power of simple groups, diameter 
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     author = {Crestani, Eleonora and Lucchini, Andrea},
     title = {The non-isolated vertices in the generating graph of a direct powers of simple groups.},
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Crestani, Eleonora; Lucchini, Andrea. The non-isolated vertices in the generating graph of a direct powers of simple groups.. Journal of Algebraic Combinatorics, Tome 37 (2013) no. 2, pp. 249-263. http://geodesic.mathdoc.fr/item/JAC_2013__37_2_a6/