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Symmetries and symmetry-breaking in arithmetic graphs
Aqsa Shah1, Imran Javaid1, Shahid Ur Rehman1
1Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.
Heliyon
|October 9, 2023
Summary
This study explores the symmetries of arithmetic graphs for composite numbers. Researchers determined the automorphism group and developed formulas for the fixing number, a measure of symmetry-breaking.
Area of Science:
- Graph Theory
- Number Theory
- Abstract Algebra
Background:
- Arithmetic graphs provide a visual representation of number-theoretic properties.
- Understanding graph symmetries is crucial in various mathematical fields.
- Symmetry-breaking concepts are essential for analyzing complex structures.
Purpose of the Study:
- To investigate the symmetries of the arithmetic graph of a composite number m.
- To analyze symmetry-breaking using the fixing number concept.
- To determine the automorphism group and derive formulas for the fixing number.
Main Methods:
- Graph traversal and analysis to determine vertex distances and degrees.
- Group theory to characterize the automorphism group of the arithmetic graph.
- Combinatorial methods to derive exact formulas for the fixing number.
Main Results:
- The automorphism group of the arithmetic graph is isomorphic to the symmetric group for .
- Exact formulas for the fixing number of were derived under specific conditions on .
- Properties like vertex distance, vertex degree, and twin classes were analyzed.
Conclusions:
- The study provides a comprehensive understanding of symmetries and symmetry-breaking in arithmetic graphs.
- The findings contribute to the structural analysis of number-theoretic graphs.
- The derived formulas offer tools for quantifying symmetry-breaking in these graphs.
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