Connectivity of Non-Coprime Graph over Dihedral Group
DOI:
https://doi.org/10.20956/y76mg453Keywords:
Non Coprime Graph, dihedral group, Eulerian Graph, hamiltonian graph, PlanarityAbstract
Let Γ be the non-coprime graph of the dihedral group of order 2n. This research investigates the connectivity properties of Γ, particularly its Eulerian, semi-Eulerian, Hamiltonian, and planar properties. The research begins by examining several properties related to the dihedral group, and then identifying the connectivity patterns that are formed in the non-coprime graph through the analysis of the order of elements in the group. These patterns then serve as the basis for determining its connectivity properties. The results show that Γ is Eulerian when n = 2k, where k ≥ 2, and semi-Eulerian if and only if n = 2p, where p is an odd prime. Furthermore, the graph is Hamiltonian for n = 2k or n = 2p, where p is an odd prime. In addition, the graph is planar if and only if n = 2 or n = 3. These results indicate that the factorization form of n strongly influences the connectivity structure and graph properties of the non-coprime graph over the dihedral group.
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