On the Inverse Problem for Some Topological Indices

Abstract

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value $\left(g\right)$ to every graph within a given family $\left(\mathcal{G}\right)$ of graphs, find some $G\in \mathcal{G}$ for which $\text{TI}\left(G\right)=g$ . It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the $Y$ -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated $F$ -index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.