Some New Results Concerning Power Graphs and Enhanced Power Graphs of Groups
Ivica Bo\v{s}njak, Roz\'alia Madar\'asz, Samir Zahirovi\'c

TL;DR
This paper explores the relationships between power graphs and enhanced power graphs of groups, proving isomorphism implications, examining perfection properties across various group orders, and characterizing specific groups with perfect enhanced graphs.
Contribution
It establishes that isomorphic power graphs imply isomorphic enhanced power graphs, and characterizes groups with perfect enhanced power graphs, including symmetric and alternating groups.
Findings
Isomorphic power graphs imply isomorphic enhanced power graphs.
Finite groups of order p^n q and p^2 q^2 have perfect enhanced power graphs.
Symmetric and alternating groups are characterized by their perfect enhanced power graphs.
Abstract
The directed power graph of a group is the simple digraph with vertex set such that if is a power of . The power graph of , denoted by , is the underlying simple graph. The enhanced power graph of is the simple graph with vertex set in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any and prime numbers and ,…
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Synthesis and Characterization of Heterocyclic Compounds
