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ON THE LAPLACIAN ENERGY OF RELATIVE
O N T H E L A P L A C I A N E N E R G Y O F R E L A T I V E K242/22
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CO - PRIME GRAPHS OF DIHEDRAL GROUPSS
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FOR ODD DEGREES LESS THAN 100
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AZMY ADAM BIN AZMY DR NORARIDA BINTI ABD RHANI
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ABSTRACTT
This research focuses on calculating the Laplacian energy of the relative coprime graphs of the dihedral groups D , D , D , and D . The
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objectives of the study are to compute the characteristic polynomials of the Laplacian matrices and calculate the corresponding Laplacian
energies. The methodology involves several key steps. Based on Sahrom (2025), the relative coprime graphs of dihedral groups for D , D , D ,
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and D are obtained. From these graphs, the Laplacian matrices are constructed followed by the computation of their characteristic
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polynomials. Finally, the Laplacian energy of the relative coprime graphs for D , D , D , and D is computed using the eigenvalues derived from
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the characteristic polynomials. The results for the Laplacian energy of the relative coprime graphs for D , D , D , and D indicate that the
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Laplacian energy increases with the size of the original group and the order of the subgroup. Subgroups associated with larger dihedral groups
tend to produce higher Laplacian energy values, while smaller groups yield lower energies. In conclusion, the findings of the Laplacian energy of
relative coprime graphs of dihedral groups for D , D , D , and D are successfully computed in this research. It is recommended that future
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research explore the Seidel energy of relative coprime graphs of these dihedral groups.
PROBLEM RESEARCH OBJECTIVESOBJECTIVES
PROBLEM STATEMENTSTATEMENT
RESEARCH
RESEARCHERS’ STUDIES GAP IN RESEARCH To determine the Laplacian matrix of the relative coprime graph of dihedral
The researchers collectively There has been limited research groups for odd degree less than 10.
highlighted important aspects on the Laplacian energy of
of dihedral group graphs, relative coprime graphs of To determine the characteristic polynomial of the Laplacian matrix of the
focusing on coprime graphs, dihedral groups for odd degrees relative coprime graph of dihedral groups for odd degree less than 10.
relative coprime graphs, and less than 10.
Laplacian energy. To compute the Laplacian energy of the relative coprime graph of dihedral
groups for odd degrees less than 10.
METHODOLOGYY
M E T H O D O L O G IMPLEMENTATION
IMPLEMENTATION
Start
Phase 1:
Determine the Laplacian matrix of
the relative coprime graph for the
dihedral groups D , D , D , and D 9
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The relative coprime graphs of D , D , D , and D are
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utilized. The graphs used in this study are adopted from
Phase 2: the works of Sahrom (2025), where the construction of the
Determine the characteristic graphs is based on Definition 3.2.1 and Definition 3.2.3.
polynomial of the Laplacian matrix.
Phase 3:
Compute the Laplacian energy for
Phase 3: Phase 1: the relative coprime graph of
Compute the Laplacian energy for Determine the Laplacian matrix dihedral groups for odd degrees
the relative coprime graph of of the relative coprime graph for less than 10.
dihedral groups for odd degrees the dihedral groups
less than 10.
End
Phase 2:
Determine the characteristic
polynomial of the Laplacian matrix.
RESULTs &
RESULTs & DISCUSSIONDISCUSSION
CONCLUSION
CONCLUSION
RECOMMENDATIONs
The Laplacian energy of RECOMMENDATIONs
relative coprime graphs varies The objective of this project has been achieved
according to order of the by applying all steps in the methodology. The
subgroups. It reveals that with result displays the Laplacian energy value that Perform a research about Seidel energy.
the increase in the order of the the researcher looked for by searching for
subgroups, the Laplacian Laplacian matrices, characteristic polynomials Perform extend the study on the Laplacian energy
energy of the relative coprime and eigenvalues. It improved knowledge of of relative coprime graphs for other groups.
graphs of the respective algebra and graph theory by analyzing the
subgroups also tends to Laplacian energy of the relative coprime graphs Extend to investigate the topological indices.
increase.
of dihedral groups for odd degrees less than 10.
