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
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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