Power graph of a finite group is always divisor graph

Abstract

The power graph is a special type undirected simple graph of a finite group G which has the group elements as its vertex set and two distinct vertices [Formula: see text] are adjacent if one is non-negative integral power of other. Vertex labeling of graph [Formula: see text] is a process of assigning integers to all its vertices subject to certain conditions. In simple words, vertex (edge) labeling is a function of all vertices (edges) to set of labels. Frequently, integers are used in labeling of vertices and edges. A graph [Formula: see text] is said to be divisor graph if all vertices of graph can be labeled with positive integers such that any two distinct vertices [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. In this research paper, we prove that every power graph of finite group is always a divisor graph, but converse is not true.

Spectral Properties of Power Graph of Dihedral Groups

“…Furthermore, the discussion on the degree formula of the power graph of some finite group can be found in [14]. Later, Takshak, et al [15] showed the new finding that if Γ G is a power graph of a finite group G, then it is a divisor graph. Kumar et al [7] have presented a complete and excellent survey of the power graph for some finite groups.…”

Topological Indices and Structural Properties of Cubic Power Graph of Dihedral Group

“…In recent years, various graphs associated with algebraic structures have been studied such as various properties of power graphs in [1][2][3][4][5], square element graphs in [6], square power graphs in [7], kth power graphs in [8], etc. A power graph of a finite group G is an undirected finite simple graph having a vertex set group G in which two distinct element vertices have an edge iff one is the power of the other.…”

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

“…In addition, Kumar et al [10] have discussed a survey of the power graph for some finite groups. The fact that a power graph is always a divisor graph is presented in [22].…”