Page 27 - programme book

AA-007
Relation Between Randic and Harmonic Energies of Commuting Graph
for Dihedral Groups

Mamika Ujianita Romdhini 1, 3, b) and Athirah Nawawi 1,2, a)

1 Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia.
2 Institute for Mathematical Research, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia.
3 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Mataram,
83125, Indonesia.

a) Corresponding author: athirah@upm.edu.my
b) mamika@unram.ac.id

Abstract. Suppose that is a group and ( ) is the center of the group . In this paper, we discuss a
specific type of graph known as the commuting graph, denoted by , whose vertex set contains all

group elements excluding central elements, \ ( ). This graph has to satisfy a condition in which
, ∈ \ ( ) where ≠ are adjacent whenever = . The number of vertices

adjacent to is denoted as , which is the degree of . The Randic and harmonic matrices of

are defined as square matrices in which ( , )-th entry are 1 and 2 if and are

+
� ∙
adjacents, respectively; otherwise, it is zero. Randic energy is the sum of the absolute eigenvalues of
the Randic matrix whereas harmonic energy is the sum of the absolute eigenvalues of the harmonic
matrix. In this paper, we compare Randic and harmonic energies focusing on the commuting graph for
dihedral group of order 2 , .
2
Keywords: Commuting graph, Randic energy, Harmonic energy, Dihedral group