Perfect codes in power graphs of finite groups

Abstract

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.

Perfect Codes in Induced Subgraph of Unit Graph Associated With Some Commutative Rings

“…Recently, there are some numbers of researches focused on perfect codes in graphs associated with groups. Ma et al [15] found the lower and upper bounds for the order of the subset H of a group G to be a perfect code in the power graph of . G In addition, they characterized the group G in which the enhanced power graph of G admits the trivial perfect codes.…”

Some properties of various graphs associated with finite groups

“…The power and commuting graphs have been also considered in many other papers, see for instance [2], [4]- [7], [10]- [18], [20,22,23], [25]- [28], [31]- [33], [41]- [44], [46]- [48], [50]- [55], [58,59,63]. In particular, in [45,Lemma 4.1], it is shown that P(G) = C(G) if and only if G is a cyclic group of prime power order, or a generalized quaternion 2-group, or a Frobenius group with kernel a cyclic p-group and complement a cyclic q-group, where p and q are distinct primes.…”

Power Graphs of Finite Groups Determined by Hosoya Properties

“…They also derived a formula of matching numbers for any finite nilpotent groups. The authors of [ 10 , 11 , 12 , 13 ] presented an overview of finite groups with enhanced power graphs that enable the formation of a perfect code. They further established all possible perfect codes of the proper reduced power graphs and gave a necessary and sufficient condition for graphs having perfect codes.…”