Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8

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• 作者：Primož Potočnik^a ; ¹ ; ^{primoz.potocnik@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author} ; Rok Požar^b ; ^{pozar.rok@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Graph ; Tetravalent ; Vertex-transitive ; Edge-transitive
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：172-183
• 全文大小：350 K

文摘

A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in Conder and Marušič (2003) [4]. This example has 10752 vertices and vertex-stabiliser isomorphic to the dihedral group of order 8. In this paper, we show that no such graphs of smaller order exist, thus answering a frequently asked question.