On Cubic Nonsymmetric Cayley Graphs

Abstract

<span>Let <img alt="" src="Edit_c92d0286-c1ee-4edf-8560-c94d9193fb89.gif" /></span><span><span></span><span><span> </span>be a connected Cayley graph of group </span></span><i><span>G</span></i><span>, then</span><span> Γ </span><span>is called normal if the right regular representation of </span><i><span>G</span></i><span> is a normal subgroup of </span><span><span></span><span>, the full automorphism group of</span></span><span> Γ</span><span>. For</span><span> the case where</span><span> </span><i><span>G</span></i><span> is a finite nonabelian simple group and</span><span> Γ </span><span>is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that</span><span> Γ</span><span> </span><span>is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley graph of nonabelian simple group aroused strong interest of people. So far such graphs which have</span><span> been</span><span> known are all normal. Then people conjecture that all of such graphs are either normal or the Cayley subset consists of involutions. In this paper we give an negative answer by two counterexamples. As far as we know these are the first examples for the non-normal cubic nonsymmetric Cayley graphs of finite nonabelian simple groups.</span><span></span

An infinite family of cubic nonnormal Cayley graphs on nonabelian simple groups

“…Until recently, connected cubic nonnormal Cayley graphs on the groups listed in Theorem 1.1 were only found for A 15 and A 31 [9]. In 2008, Feng, Lu and Xu asked the following question in their survey paper [4] on normality of Cayley graphs.…”