In 1992, H. Zhang (J. Graph Theory 16, 1-5), using the classification of finite simple groups, gave an algebraic characterisation of self-complementary symmetric graphs. Yet, from this characterisation it does not follow whether such graphs, other than the well-known Paley graphs, exist. In this paper we give a full description of self-complementary symmetric graphs and their automorphism groups. In particular, we prove that apart from the Paley graphs there is another infinite family of self-complementary symmetric graphs and, in addition, one more graph not belonging to any of these families. We obtain this by investigating automorphism groups of graphs and applying classification results on primitive permutation, groups of low rank. We prove also that the automorphism group of a self-complementary symmetric graph is permutation isomorphic to a subgroup of A GammaL(1)(p(r)) with three exceptions, when it can be presented as a subgroup of A GammaL(2)(p(r)). (C) 2001 Academic Press. [References: 23]
展开▼
机译:1992年,H。Zhang(J. Graph Theory 16,1-5)使用有限简单组的分类,给出了自互补对称图的代数表征。然而,根据这种表征,并不能确定除了众所周知的Paley图之外,是否还存在这样的图。在本文中,我们对自互补对称图及其自同构群进行了完整描述。特别是,我们证明了,除了Paley图之外,还有另一个无限的自互补对称图族,此外,还有一个不属于这些族中的任何一个的图。我们通过研究图的自同构组并将分类结果应用于原始置换(低秩组)来获得此结果。我们还证明,自互补对称图的自同构群可以置换为A GammaL(1)(p(r))的一个子群,但可以作为A GammaL(2)的一个子群出现三个例外(p(r))。 (C)2001学术出版社。 [参考:23]
展开▼