On Bounding the Diameter of the Commuting Graph of a Group
On Bounding the Diameter of the Commuting Graph of a Group
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关于群通勤图的直径有界
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
Aedan Pope
中科院分区：
文献类型：
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作者：
Michael Giudici;Aedan Pope
群G$的交换图是简单无向图，其顶点是G$的非中心元素，且两个不同的顶点相邻当且仅当它们交换。Jafarzadeh和Iranmanesh证明了有限群的交换图的直径存在一个泛上界，当交换图是连通的时.在本文中，我们确定的上界的交换图的直径为某些类的群体，以排除他们作为可能的反例，这一猜想。我们还给出了一个平凡中心直径为6的群的无限族的例子，以前已知的无限族的最大直径为5，对于$S_n$。
The commuting graph of a group $G$ is the simple undirected graph whose vertices are the non-central elements of $G$ and two distinct vertices are adjacent if and only if they commute. It is conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with trivial centre and diameter 6, the previously largest known diameter for an infinite family was 5 for $S_n$.