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Mathematics and Statistics Vol. 13(3), pp. 175 - 179
DOI: 10.13189/ms.2025.130306
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On Energy of Prime Coprime Graph for Dihedral Groups

Mamika Ujianita Romdhini ^1,*, Abdurahim ¹, Faisal Al-Sharqi ², Syamsul Bahri ¹, Andika Ellena Saufika Hakim Maharani ^1
1 Department of Mathematics, Faculty of Mathematics and Natural Science, University of Mataram, Mataram 83125, Indonesia
² Department of Mathematics, Faculty of Education for Pure Sciences, University of Anbar, Ramadi, Anbar, Iraq

ABSTRACT

The study of graphs associated with algebraic structures has become an influential approach in understanding both the combinatorial and algebraic properties of groups. In this paper, we focus on a special class of graphs known as prime coprime graphs constructed from the dihedral groups , a family of non-abelian finite groups of order . The prime coprime graph, denoted , is defined with the vertex set consisting of the non-identity elements of , where two distinct vertices are adjacent whenever the greatest common divisor of their orders is a prime number. The primary aim of this research is to examine the spectral properties of , with a focus on the graph energy, defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. We construct the adjacency matrix of for cases where is either a power of a prime or a product of two distinct primes. Through direct computation and spectral analysis, we determine the eigenvalues and calculate the energy of the graph. Additionally, we explore the relationship between the energy and the spectral radius. Our findings provide new insights into the spectral behavior of non-commutative algebraic graphs and contribute to the broader theory of graph energies. While the study is restricted to specific cases of , it opens pathways for generalizations to other classes of groups. These results may have further implications in algebraic combinatorics, coding theory, and the spectral classification of finite groups.

KEYWORDS
Adjacency Matrix, Energy of A Graph, Spectral Radius, Prime Coprime Graph, Dihedral Group

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mamika Ujianita Romdhini , Abdurahim , Faisal Al-Sharqi , Syamsul Bahri , Andika Ellena Saufika Hakim Maharani , "On Energy of Prime Coprime Graph for Dihedral Groups," Mathematics and Statistics, Vol. 13, No. 3, pp. 175 - 179, 2025. DOI: 10.13189/ms.2025.130306.

(b). APA Format:
Mamika Ujianita Romdhini , Abdurahim , Faisal Al-Sharqi , Syamsul Bahri , Andika Ellena Saufika Hakim Maharani (2025). On Energy of Prime Coprime Graph for Dihedral Groups. Mathematics and Statistics, 13(3), 175 - 179. DOI: 10.13189/ms.2025.130306.