# Perfect codes in power graphs of finite groups

Open Mathematics (2017)

• Volume: 15, Issue: 1, page 1440-1449
• ISSN: 2391-5455

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## Abstract

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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.

## How to cite

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Xuanlong Ma, et al. "Perfect codes in power graphs of finite groups." Open Mathematics 15.1 (2017): 1440-1449. <http://eudml.org/doc/288446>.

@article{XuanlongMa2017,
abstract = {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.},
author = {Xuanlong Ma, Ruiqin Fu, Xuefei Lu, Mengxia Guo, Zhiqin Zhao},
journal = {Open Mathematics},
keywords = {Power graph; Enhanced power graph; Finite group; Perfect code; Total perfect code},
language = {eng},
number = {1},
pages = {1440-1449},
title = {Perfect codes in power graphs of finite groups},
url = {http://eudml.org/doc/288446},
volume = {15},
year = {2017},
}

TY - JOUR
AU - Xuanlong Ma
AU - Ruiqin Fu
AU - Xuefei Lu
AU - Mengxia Guo
AU - Zhiqin Zhao
TI - Perfect codes in power graphs of finite groups
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 1440
EP - 1449
AB - 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.
LA - eng
KW - Power graph; Enhanced power graph; Finite group; Perfect code; Total perfect code
UR - http://eudml.org/doc/288446
ER -

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