• Interesting molecular representations are associated with topological indices based on degree and neighborhood degree sum. • An application of topological indices for the study of newly synthesized carbon allotropes, specifically δ -graphene, δ -graphyne and δ -graphdiyne are introduced. • The complexity and characteristics of carbon allotropes are explored through Renyi entropy based generalized fractal dimensions, which are newly defined using specific key types of topological indices. • GFD values of carbon allotrope structures are determined using the multifractal formalism, which is based on Renyi entropy and topological indices, to investigate the complexity of these structures and its self-similar properties. 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 provide topological indices based on degree and neighborhood degree sum that serve as effective molecular descriptors for linking structural features with physicochemical properties. Our current study demonstrates the effectiveness of topological indices in the investigation of the newly synthesized carbon allotropes, namely δ -graphene, δ -graphyne and δ -graphdiyne. Furthermore, the complexity and information of carbon allotropes can be discussed in terms of Generalized Fractal Dimensions (GFD), which are newly constructed based on Renyi entropy using certain important types of topological indices. GFD analysis is increasingly important for characterizing structural properties and quantifying the complexity of chemical graphs. This research employs the edge partitioning approach to explore multiple topological indices based on degree and neighborhood degree sum for these networks. We have calculated GFD values of these structures using the GFD method based on Renyi entropy through topological indices. The calculated GFD values are confined to a particular range and demonstrate a clear monotonic trend with the iteration parameters, indicating increasing structural complexity. These obtained GFD values are then compared both graphically and in tabular form to assess the complexity of the given carbon allotropes.
Yogalakshmi et al. (Sun,) studied this question.