Sains Malaysiana 51(12)(2022): 4145-4151

http://doi.org/10.17576/jsm-2022-5112-21

 

Maximum and Minimum Degree Energy of Commuting Graph for Dihedral Groups

(Tenaga Darjah Maksimum dan Minimum bagi Graf Kalis Tukar Tertib bagi Kumpulan Dwihedron)

 

MAMIKA UJIANITA ROMDHINI1,3 & ATHIRAH NAWAWI1,2*

1Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia
2Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

3Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Mataram, 83125, Indonesia

 

Received: 14 April 2022/Accepted: 29 August 2022

 

Abstract

If  is a finite group and  is the centre of , then the commuting graph for , denoted by , has as its vertices set with two distinct vertices  and  are adjacent if . The degree of the vertex  of , denoted by , is the number of vertices adjacent to . The maximum (or minimum) degree matrix of  is a square matrix whose -th entry is    whenever  and  are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of  for dihedral groups of order ,  by using the absolute eigenvalues of the corresponding maximum degree matrices ( ) and minimum degree matrices ( ).Here, the comparison of maximum and minimum degree energy of  for is discussed by considering odd and even  cases. The result shows that for each case, both energies are non-negative even integers and always equal.

 

Keywords: Commuting graph; degree of vertex; dihedral group; energy of a graph

                                                                                          

Abstrak

Jika  adalah suatu kumpulan terhingga dan  adalah pusat bagi , maka graf kalis tukar tertib bagi , ditatatandakan dengan , mempunyai sebagai set bucunya dengan dua bucu berbeza  dan  adalah bersebelahan jika . Darjah bucu  dalam , ditatatandakan dengan , adalah bilangan bucu bersebelahan dengan . Matriks darjah maksimum (atau minimum) bagi  ialah matriks segiempat sama yang mana unsur ke-  adalah    apabila  dan  bersebelahan, jika tidak, ia adalah sifar. Kajian ini mengemukakan tenaga darjah maksimum dan minimum  bagi kumpulan dwihedron berperingkat ,  dengan menggunakan nilai eigen mutlak bagi matriks darjah maksimum ( ) dan matriks darjah minimum ( ) yang sepadan. Di sini, perbandingan tenaga darjah maksimum dan minimum  bagi  dibincangkan dengan mempertimbangkan kes   ganjil dan genap. Hasilnya menunjukkan bahawa bagi setiap kes, kedua-dua tenaga adalah integer genap bukan negatif dan sentiasa sama.

 

Kata kunci: Darjah bucu; graf kalis tukar tertib; kumpulan dwihedron; tenaga graf

 

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*Corresponding author; email: athirah@upm.edu.my

 

 

 

 

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