Abstract
The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.
Highlights
Various techniques have been used by researchers in investigating the properties of a group as well as classifying it according to its properties, which happen to be one of theM
[11] Vahidi and Talebi obtained some invariants of non-commuting graphs on dihedral groups and generalized quaternion groups, which include its independent number, clique number and minimum size of the vertex cover of the graph, in their paper, they did not find the chromatic number of this graph, which has been obtained a year later by Tamizh et al . [9]
We give the formal definition of the order product prime graph and the general presentation for its connectivity, completeness, regularity and planarity on dihedral group and cyclic group, which help in obtaining its diameter, girth, independent number, and the clique number
Summary
Various techniques have been used by researchers in investigating the properties of a group as well as classifying it according to its properties, which happen to be one of the. One of the techniques found to be useful is by defining graph to the groups and investigate its properties using the corresponding geometric structure. In [6], where they investigated the connectivity, regularity and planarity of the graph and concurrently, give the numerical invariants of the graph which happen to be the improvement of the result given for non-commuting graphs. We defined order product prime graph of finite groups, classify groups in terms of the properties of the graph as one of connected, complete, regular, planar and obtained some of its invariants such as independent number, clique number, girth and diameter. We investigate the nilpotency status of dihedral group using the connectivity for the order product prime graphs on the dihedral group
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