Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A graph ΓΓ is said to be symmetric   if its automorphism group Aut(Γ)Aut(Γ) is transitive on the arc set of ΓΓ. Let GG be a finite non-abelian simple group and let ΓΓ be a connected pentavalent symmetric graph with G≤Aut(Γ)G≤Aut(Γ). ,  show that if GG is transitive on the vertex set of ΓΓ, then either G⊴Aut(Γ)G⊴Aut(Γ) or Aut(Γ)Aut(Γ) contains a non-abelian simple normal subgroup TT such that G≤TG≤T and (G,T)(G,T) is one of 5858 possible pairs of non-abelian simple groups. In particular, if GG is transitive on the arc set of ΓΓ, then (G,T)(G,T) is one of 17 possible pairs, and if GG is regular on the vertex set of ΓΓ, then (G,T)(G,T) is one of 1313 possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, et al. (2011).
['Jiali Du', 'Yan-Quan Feng', 'Jin-Xin Zhou']
I am writing a research paper on a idea Pentavalent symmetric graphs admitting vertex-transitive non-abelian simple groups
[{"from": "human", "value": "I am writing a research paper on a idea Pentavalent symmetric graphs admitting vertex-transitive non-abelian simple groups"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A graph \u0393\u0393 is said to be symmetric \u00a0 if its automorphism group Aut(\u0393)Aut(\u0393) is transitive on the arc set of \u0393\u0393. Let GG be a finite non-abelian simple group and let \u0393\u0393 be a connected pentavalent symmetric graph with G\u2264Aut(\u0393)G\u2264Aut(\u0393). ,  show that if GG is transitive on the vertex set of \u0393\u0393, then either G\u22b4Aut(\u0393)G\u22b4Aut(\u0393) or Aut(\u0393)Aut(\u0393) contains a non-abelian simple normal subgroup TT such that G\u2264TG\u2264T and (G,T)(G,T) is one of 5858 possible pairs of non-abelian simple groups. In particular, if GG is transitive on the arc set of \u0393\u0393, then (G,T)(G,T) is one of 17 possible pairs, and if GG is regular on the vertex set of \u0393\u0393, then (G,T)(G,T) is one of 1313 possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, et al. (2011)."}]