TL;DR
This paper studies the structure of the commuting graph in finite soluble groups with trivial centers, establishing bounds on its diameter and providing examples of groups with specific graph properties.
Contribution
It proves that the diameter of the commuting graph is at most 8 or the graph is disconnected, and constructs examples of groups with diameter 8.
Findings
Diameter of commuting graph is at most 8 or disconnected
Examples of soluble groups with diameter exactly 8
Insights into the structure of commuting graphs in soluble groups
Abstract
The commuting graph of a finite soluble group with trivial centre is investigated. It is shown that the diameter of such a graph is at most 8 or the graph is disconnected. Examples of soluble groups with trivial centre and commuting graph of diameter 8 are provided.
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