当前位置: X-MOL 学术J. Algebraic Comb. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On groups all of whose Haar graphs are Cayley graphs
Journal of Algebraic Combinatorics ( IF 0.6 ) Pub Date : 2019-06-10 , DOI: 10.1007/s10801-019-00894-7
Yan-Quan Feng , István Kovács , Da-Wei Yang

A Cayley graph of a group H is a finite simple graph \(\Gamma \) such that \(\mathrm{Aut}(\Gamma )\) contains a subgroup isomorphic to H acting regularly on \(V(\Gamma ),\) while a Haar graph of H is a finite simple bipartite graph \(\Sigma \) such that \(\mathrm{Aut}(\Sigma )\) contains a subgroup isomorphic to H acting semiregularly on \(V(\Sigma )\) and the H-orbits are equal to the bipartite sets of \(\Sigma \). A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that \(D_6, \, D_8, \, D_{10}\) and \(Q_8\) are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph.

中文翻译:

在所有其Haar图为Cayley图的组上

H组的Cayley图是有限的简单图\(\ Gamma \),因此\(\ mathrm {Aut}(\ Gamma)\)包含与H规则同构地作用于\(V(\ Gamma)的子集, \),H的Haar图是有限的简单二部图\(\ Sigma \),因此\(\ mathrm {Aut}(\ Sigma)\)包含与H同构的半同构作用于\ {V(\ Sigma )\)H轨道等于\(\ Sigma \)的二分集。Cayley图恰好是二分时才是Haar图,但是没有简单的条件可让Haar图成为Cayley图。在本文中,我们证明\(D_6,\,D_8,\,D_ {10} \)\(Q_8 \)是仅有的有限内部阿贝尔群,它们的Haar图都是Cayley图。(如果一个组是非阿贝尔的,则称为内部阿贝尔,但其所有适当的子组都是阿贝尔的。)然后将使用此结果来推导每个不可解的组都承认一个非凯利Haar图。
更新日期:2019-06-10
down
wechat
bug