On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Gamma = Cay(G,S) is said to be normal if G is normal in Aut Gamma. In this paper, we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Gamma of finite nonabelian simple groups G, where the vertex stabilizer Av is soluble for A=Aut Gamma and v is an element of V Gamma. We prove that either Gamma is normal or G=A(5), A(10), A(54), A(274), A(549) or A(1099). Further, 11-valent symmetric nonnormal Cayley graphs of A(5), A(54) and A(274) are constructed. This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind (of valency 11) was constructed by Fang, Ma and Wang in 2011.