Perfect codes in power graphs of finite groups

Open Mathematics

Guo Mengxia¹  Ma Xuanlong²  Lu Xuefei²  Fu Ruiqin²  Zhao Zhiqin² 

[1] Henan Polytechnic University Hebi Institute of Engineering and Technology, Hebi Polytechnic, 458030, Hebi, China;School of Science, Xi’an Shiyou University, 710065, Xi’an, China;

关键词: power graph;    enhanced power graph;    finite group;    perfect code;    total perfect code;    05c25;    05c69;    94b99;   

DOI  :  10.1515/math-2017-0123

来源: DOAJ

【 摘 要 】

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

【 授权许可】

Unknown