On the power graphs of certain finite groups

ABSTRACT

The power graph of a group Ω is a graph, whose node set is Ω and two distinct elements are adjacent if and only if one is an integral power of the other. A metric dimension of a graph Γ, denoted by ψ(Γ) is the minimum cardinality of the resolving set of Γ. In this context, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraphs of the power graphs of certain finite non-abelian groups. As a consequence, we figure out the metric dimension and resolving polynomial of power graphs for dihedral and generalized quaternion groups by using neighbourhood and twin sets.

Keywords:

• Power graphs
• finite groups
• distant properties
• metric dimension
• resolving polynomial

AMS Subject Classifications:

• 05C07
• 05C12
• 05C25
• 05C50

Acknowledgments

The authors are indebted to the anonymous reviewers for their careful reading of the paper and for their valuable suggestions which greatly improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11971376].