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
2
Institute 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

 

Diserahkan: 14 April 2022/Diterima: 29 Ogos 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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*Pengarang untuk surat-menyurat; email: athirah@upm.edu.my

 

 

 

   

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