The intrinsic properties of carbon nanosheets derived from their basic molecular structure through a self-similar pattern have attracted much interest among researchers. Graph-theoretical methods are used to identify certain molecular descriptors known as topological indices, which are highly useful in connecting molecules to their physical attributes. Several chemical characteristics have been correlated with degree and neighborhood degree sum-based topological indices, which have been investigated extensively. Our current research establishes the use of topological indices in studying the newly synthesized carbon allotropes $\delta$-graphene, $\delta$-graphyne and $\delta$-graphdiyne. Furthermore, the complexity and information of carbon allotropes can be discussed in terms of Generalized Fractal Dimensions $(\mathrm{GFD})$, which are newly constructed based on Renyi entropy using some types of topological indices. The study of GFD indices of graphs is gaining importance as a measure of the complexity of basic coupling and as a tool for characterizing structural properties. We have estimated various topological indices, including graph-based GFD values of these structures obtained using the GFD method based on Renyi entropy.