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September 10, 2025Open Access

Spectral Properties of Coprime Graphs for Dihedral Groups

Key Points

The analysis determines the characteristic polynomial for coprime graphs in dihedral groups, revealing key spectral properties.
Findings on the energy of the graph indicate a relationship with the orders of the group elements.
The study focuses on dihedral group of order 2n, where n is a power of a prime number to highlight its structural importance.
Understanding the spectral properties enhances knowledge of group theory and can have implications for algebraic graph theory.

Abstract

For a finite group G, the coprime graph ΓG of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in ΓG are adjacent if and only if gcd(|u|, |v|) = 1, where |u| is the order of u. This study analyzes the characteristic polynomial of matrices for the dihedral group of order 2n, where n is a power of a prime number. In addition, this paper examines the characteristic polynomial of the matrices for a power of a prime number n. The energy of the graph is also obtained.

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Cite This Study

Romdhini et al. (Fri,) studied this question.

synapsesocial.com/papers/68c1a77a54b1d3bfb60e0b9c https://doi.org/https://doi.org/10.29020/nybg.ejpam.v18i3.6319

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