Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups

2017/02/19 by Bo Ling, Ling, Bo, Ben Gong Lou +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1702.05754

10 pages

arxiv created 2017/02/19 · arxiv updated 2017/02/21

Abstract

A Cayley graph \Ga=\Cay(G,S) is said to be normal if G is normal in \Aut\Ga. The concept of normal Cayley graphs was first proposed by M.Y.Xu in [Discrete Math. 182, 309-319, 1998] and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we investigate the normality problem of the connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups. We prove that all such graphs \Ga are either normal or G=\A₃₉ or \A₇₉. Further, a connected arc-transitive pentavalent Cayley graph on \A₇₉ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.

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