Seminar on “The Hyper-Zagreb Index of the Coprime Graph of Subgroups of a Cyclic Group Z_n”
A seminar entitled “The Hyper-Zagreb Index of the Coprime Graph of Subgroups of a Cyclic Group Z_n” was presented by Bushra Abdulgaphur in the Department of Mathematics, College of Education, on 24 May 2026.
This seminar explores the intersection of algebraic group theory and graph theory by analyzing the structural properties of the coprime graph of Z_n. The vertex set consists of all nontrivial proper subgroups of the cyclic group Z_n, where two distinct vertices are adjacent if and only if their orders are relatively prime. The primary objective of this study is to provide a comprehensive topological classification of the Hyper-Zagreb Index (HZI) based on the prime factorization of n.
Key Finding and Results
he structural complexity of these graphs is systematically categorized into distinct number-theoretic cases based on the arithmetic nature of n
Prime Power Case (n=p^k): The graph is identified as a null graph with no edges, resulting in an index of zero.
Two-Prime Power Case( n=p^k q^r): The general algebraic formula derived for this structure yields an index of kr(k+r)^2
Three Distinct Primes (n=pqr): Due to its square-free triple factor structure, the Hyper-Zagreb Index always results in a constant value of 156.
Composite Case (n=p^2 qr):Detailed degree-sum analysis for this multipartite structure yields a constant index value of 550.
Ultimately, these results establish a significant mathematical bridge linking algebraic group invariants with the topological descriptors of graphs derived from geometric and algebraic structures
