Geodesic
An Adjacency Criterion for the Prime Graph of a Finite Simple Group
Algebra i logika, Tome 44 (2005) no. 6, pp. 682-725

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For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.
Keywords: finite group, finite simple group, group of Lie type, spectrum of a finite group, recognition by spectrum, prime graph of a finite group, independence number of a prime graph, 2-independence number of a prime graph. 
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     author = {A. V. Vasil'ev and E. P. Vdovin},
     title = {An {Adjacency} {Criterion} for the {Prime} {Graph} of {a~Finite} {Simple} {Group}},
     journal = {Algebra i logika},
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     url = {http://geodesic.mathdoc.fr/item/AL_2005_44_6_a2/}
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A. V. Vasil'ev; E. P. Vdovin. An Adjacency Criterion for the Prime Graph of a Finite Simple Group. Algebra i logika, Tome 44 (2005) no. 6, pp. 682-725. http://geodesic.mathdoc.fr/item/AL_2005_44_6_a2/