On the power graph of a finite group
Min Feng, Xuanlong Ma, Kaishun Wang

TL;DR
This paper investigates the structural properties of the power graph of finite groups, proving it is a perfect graph with a complete core, characterizing its structure via cyclic subgroups, and deriving a formula for its metric dimension.
Contribution
It establishes the perfectness of the power graph, characterizes its structure using cyclic subgroups, and provides a closed formula for its metric dimension, including specific computations for cyclic groups.
Findings
Power graph has a transitive orientation and is a perfect graph.
The core of the power graph is a complete graph.
A closed formula for the metric dimension of the power graph is derived.
Abstract
The power graph of a finite group is the graph with the vertex set , where two elements are adjacent if one is a power of the other. We first show that has an transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups (under usual inclusion) to characterise the structure of . Finally, the closed formula for the metric dimension of is established. As an application, we compute the metric dimension of the power graph of a cyclic group.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · graph theory and CDMA systems · Advanced Graph Theory Research
