On the Exact Values of HZ-Index for the Graphs under Operations

Abstract

Topological index (TI) is a function from the set of graphs to the set of real numbers that associates a unique real number to each graph, and two graphs necessarily have the same value of the TI if these are structurally isomorphic. In this note, we compute the $\text{HZ}-\text{index}$ of the four generalized sum graphs in the form of the various Zagreb indices of their factor graphs. These graphs are obtained by the strong product of the graphs $G$ and $D_k\left(G\right)$ , where $D_k\in \left\{S_k,R_k,Q_k,T_k\right\}$ represents the four generalized subdivision-related operations for the integral value of $k\ge 1$ and $D_k\left(G\right)$ is a graph that is obtained by applying $D_k$ on $G$ . At the end, as an illustration, we compute the $\text{HZ}-\text{index}$ of the generalized sum graphs for exactly $k=1$ and compare the obtained results.