幂图为余图的有限群

Finite Groups Whose Power Graphs are Cographs

设G为有限群,定义在G上的幂图以G为顶点集,其中两个不同的顶点相邻当且仅当一个能表示成另外一个的方幂.如果一个图没有同构于四个顶点路的诱导子图,则称该图为余图.最近,Peter提出了分类幂图为余图的有限群问题,本文从群的极大循环子群和元素的中心化子出发,刻画了幂图为余图的有限群.作为应用,本文也分类了幂图为余图的几类有限群,如幂零群、二面体群、广义四元素群和对称群等.

Let G be a finite group.The power graph of G is a graph whose vertex set is G,where two distinct vertices are adjacent if and only if one is a power of the other.If a graph does not contain the four-vertex path as an induced subgraph,then this graph is called a cograph.Recently,Peter J.Cameron put forward this question:Classify the finite groups whose power graphs are cographs.In view of maximal cyclic subgroups and centralizers of elements in a group,we characterize all finite groups whose power graphs are cographs.As applications,we also classify some classes of finite groups whose power graphs are cographs,such as,nilpotent groups,dihedral groups,generalized quaternion groups,and symmetric groups and so on.