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The number of Hamiltonian cycles in groups of Symmetry
- 1 Nanjing Jinling High School
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Abstract
Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.
Keywords
Symmetric Groups, Cayley Graphs, Hamiltonian Cycles
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[2]. Tripi, Anna, “Cayley Graphs of Groups and Their Applications” (2017). MSU Graduate Theses. 3133. https://bearworks.missouristate.edu/theses/313
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Cite this article
Jiang,Z. (2024). The number of Hamiltonian cycles in groups of Symmetry. Theoretical and Natural Science,43,1-5.
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Volume title: Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
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