A Novel Approach to Topological Indices of the Power Graph Associated with the Quaternion of a Certain Order

Abstract

The power graph of a group $G$, denoted $\Gamma_G$, is defined as a graph whose vertices correspond to the elements of $G$. Two distinct vertices $a,b\in G$ re adjacent if and only if there exists a positive integer $m$ such that $a^m=b$ or $b^m=a$ This study investigates a new approach for computing the topological indices of the power graph associated with the generalized quaternion group $Q_{4n}$, where $n=p^k$, $p$ is prime number and $k\in\mathbb{Z}$. The proposed method relies on structural properties of the power graph, particularly its diameter, allowing the indices to be determined directly from the number of vertices and edges. This enables a more direct and simplified calculation process compared to conventional methods, which often require more detailed combinatorial analysis or case-based enumeration. The results obtained using this new approach match exactly with those derived through traditional techniques. This confirms that the proposed method is not only valid but also efficient in computing topological indices for the power graph of the generalized quaternion group. Overall, the findings demonstrate that structural characteristics of the power graph can be effectively utilized to streamline the computation of its associated indices.