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ON THE LAPLACIAN ENERGY OF RELATIVE CO -PRIME GRAPHSS
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OF DIHEDRAL GROUPS WITH EVEN DEGREES UP TO 100
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MUHAMAD FIRDAUS BIN ABDULLAH í 2022882408
SUPERVISORz DR NORARIDA BINTI ABD RHANI
FACULTY OF COMPUTER AND MATHEMATICAL SCIENCES
ABSTRACT OBJECTIVES
Thiv revearch analev |he Laplacian energ of rela|ie co-prime graphv for dihedral gropv To de|ermine |he Laplacian ma|ri of |he rela|ie co-prime
D , D , D , and D . I| inolev conv|rc|ing Laplacian ma|ricev from graphv baved on Mohd graph of dihedral gropv for een degreev p |o ţŢ
Ū
ţŢ
Ŧ
Ũ
Sbri (ŤŢŤŧ) ork, calcla|ing |heir charac|eriv|ic polnomialv, and deriing eigenalev |o To de|ermine |he charac|eriv|ic polnomial of |he Laplacian
comp|e Laplacian energ. The revl|v vho |ha| Laplacian energ ariev i|h |he ma|ri of |he rela|ie co-prime graph of dihedral gropv for
complei| and order of |he gropv higher for more comple v|rc|rev. The v|d een degreev p |o ţŢ
con|rib|ev |o vpec|ral graph |heor and encoragev fr|her eplora|ion of vimilar grop To comp|e |he Laplacian energ of |he rela|ie co-prime
v|rc|rev for deeper vpec|ral invigh|v. graph of dihedral gropv for een degreev p |o ţŢ
PROBLEM STATEMENT IMPLEMENTATION
The v|d of Laplacian energ iv a ke |ool in
analing graph vpec|ra and hav been idel P h a s e 1
Phase 1
eplored for ariov graphv and gropv. Hoeer, Determine the Laplacian matrix of the relative
|he Laplacian energ of rela|ie co-prime graphv co-prime graph for dihedral groups D , D6, D 8
4
and D
of dihedral gropv, par|iclarl for een degreev 10
p |o ţŢ, remainv ndereplored. Thiv revearch
aimv |o fill |ha| gap b inev|iga|ing |he vpec|ral
proper|iev of |heve graphv, |hereb con|rib|ing |o
|he broader field of algebraic graph |heor.
METHODOLOGY
The rela|ie co-prime graphv of D ,
Ŧ
Start D , D , and D are |ilied. The
Ū
Ũ
ţŢ
graphv ved in |hiv v|d are
adop|ed from |he orkv of Mohd
Sbri (ŤŢŤŧ) here |he
e
s
a
h
Phase 11
P
Determine the Laplacian matrix of the relative conv|rc|ion of |he graphv iv baved
co-prime graph for dihedral groups D , D6, D 8 on |he Defini|ion ť.Ť.ţ and
4
and D 10 Defini|ion ť.Ť.ť. Determine the characteristic polynomial
s
h
a
Phase 22
P
e
of the Laplacian matrix
Phase 22
P
h
e
a
s
Determine the characteristic polynomial of
the Laplacian matrix
a
h
P
Phase 33
s
e
Compute the Laplacian energy for the
relative co-prime graph of dihedral groups P h a s e
Phase 33
of even degrees up to 10 Compute the Laplacian energy for the relative
co-prime graph of dihedral groups of even
degrees up to 10
End
RESULTS & DISCUSSION
CONCLUSION RECOMMENDATIONS
The conclvion highligh|v |ha| |hiv revearch iv |he firv| F|re revearch vhold inev|iga|e Seidel
|o eplore |he Laplacian energ of rela|ie co-prime energ in rela|ie co-prime graphv of dihedral
The Laplacian energ of rela|ie co-prime graphv graphv for dihedral gropv i|h een degreev p |o ţŢ. gropv |o compare i| i|h Laplacian energ.
iv inflenced b |he v|rc|re and reglari| of |he I| vmmariev |he v|d'v ke v|epv from graph
vbgropv. Av |he order of a vbgrop increavev, conv|rc|ion and Laplacian ma|ri formla|ion |o |he The v|d can be e|ended |o o|her grop
|he graph |picall becomev more vmme|ric and calcla|ion of Laplacian energiev and emphaviev i|v |pev, vch av vmme|ric and qa|ernion
in|erconnec|ed, hich con|rib|ev |o a more con|rib|ion |o algebraic graph |heor. The findingv gropv, |o obvere aria|ionv in graph
v|able or elea|ed Laplacian energ. Thiv indica|ev offer alable invigh|v in|o |he rela|ionvhip be|een v|rc|re and energ.
grop v|rc|re and graph energ. The v|d alvo
|ha| higher-order vbgropv of|en prodce vggev|v direc|ionv for f|re revearch, inclding Fr|her revearch vhold eplore |opological
graphv i|h richer v|rc|ral pa||ernv, leading |o eploring larger dihedral gropv, comparing differen| indicev |o gain deeper invigh| in|o |he
higher or more conviv|en| energ alev. |pev of graph energiev, and appling |he me|hodv |o v|rc|ral proper|iev of rela|ie co-prime
o|her algebraic v|rc|rev. graphv.
