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.
@article{bwmeta1.element.doi-10_1515_math-2017-0123,
author = {Xuanlong Ma and Ruiqin Fu and Xuefei Lu and Mengxia Guo and Zhiqin Zhao},
title = {Perfect codes in power graphs of finite groups},
journal = {Open Mathematics},
volume = {15},
year = {2017},
pages = {1440-1449},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2017-0123}
}
Xuanlong Ma; Ruiqin Fu; Xuefei Lu; Mengxia Guo; Zhiqin Zhao. Perfect codes in power graphs of finite groups. Open Mathematics, Tome 15 (2017) pp. 1440-1449. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2017-0123/