Page 19 - POSTER FYP MAC-OGOS 2025

K242/19

SUPERVISOR: DR FADILA NORMAHIA ABD MANAF

AHMAD A’ZIEMULLAH UMAR BIN AZAILANI | 2022475062

PROBLEM STATEMENT

ABSTRACT

Lack of Research on Topological Indices in n-th coprime

This project explores the use of topological indices, which are the first Graphs of Q₈

Zagreb, second Zagreb, and Wiener indices, on the n-th coprime graph Need for General Expressions

of the Quaternion group Q₈, a non-abelian group of order 8. The n-th Structural Understanding and Theoretical Contribution

coprime graph is constructed using the greatest common divisor

(GCD) of element orders to form a 0-1 Table. The indices are analyzed

across different values of n, revealing structures such as star graphs, OBJECTIVES

complete tripartite graphs, and complete graphs. These results

enhance the understanding of non-abelian group graphs and support To construct the n-th coprime graph of the Quaternion group, Q₈

further study in algebraic graph theory. To determine the first Zagreb index for the n-th coprime graph of Q₈

To compute the second Zagreb index for the n-th coprime graph of Q₈.

To find the Wiener index for the n-th coprime graph of Q₈

METHOD & Cayley Table of Q₈ xⁿ Table for all x in element Q₈

IMPLEMENTATION xⁿ Table developed 0-1 Table created by using GCD

for value n = 1,3,5, and 7 equal exactly as

n = 1,2,3,4,5,6,7,8 elements of Q₈
PHASE 1 n = 2 and 6 consists of e and a²

n = 4 and 8 are equal to identity

0-1 Table for n = 1,3,5 and 7 0-1 Table for n = 2 and 6 0-1 Table for n = 4 and 8

PHASE 2 PHASE 3 PHASE 4

For the first Zagreb index: For the second Zagreb index: For the first Zagreb index: For the second Zagreb index: For the first Zagreb index: For the second Zagreb index:

For the Wiener index:

For the Wiener index: For the Wiener index:

The n-th coprime graph The first Zagreb index: The second Zagreb index: The Wiener index:
RESULT &

DISCUSSION

CONCLUSION

RECOMMENDATION

Topological Indices Successfully Applied.

Graph Structure Reflects Index Behavior. Explore additional topological indices

-Increases in the Zagreb indices correspond to larger, more connected, and more complex graphs, such as Szeged, Harary, and other indices.

indicating stronger physical and chemical properties.

Wiener Index Shows Inverse Trend. Extend to other non-abelian groups for

-A decrease in the Wiener index highlights its sensitivity to vertex distances, constructing the n-th coprime graphs.

with lower values suggesting reduced structural stability.