# Perfect codes in power graphs of finite groups

Xuanlong Ma; Ruiqin Fu; Xuefei Lu; Mengxia Guo; Zhiqin Zhao

Open Mathematics (2017)

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

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topXuanlong 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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