Abstract:
Chemical graph theory plays a pivotal role in mathematical chemistry by representing
chemical structures as graphs, with vertices denoting atoms and edges denoting chemical
bonds. Topological indices, numerical invariants derived from such graphs, have been
widely employed in quantitative structure property relationship (𝑄𝑆𝑃𝑅) and quantitative
structure activity relationship (𝑄𝑆𝐴𝑅) studies. These indices correlate molecular structure
with physicochemical and biological properties and have become crucial tools in drug
design. Supramolecular chemistry, which studies entities formed bymolecular self-assembly
through non-covalent interactions, offers an exciting avenue for designing complex
molecular architectures. In this work, we investigate the supramolecular structure of
Fuchsine (C₂₀H₁₉N₃HCl), a magenta dye of significant microbiological and histological
importance. We construct a supramolecular sheet, denoted [𝑚, 𝑛], comprising 𝑚 × 𝑛 units
of Fuchsine molecules. The corresponding chemical graph is simple, connected, and finite,
consisting of 38𝑚𝑛 + 𝑚 + 𝑛 vertices and 42𝑚𝑛 edges, which are further classified by the
Revan degrees of their end vertices. We derive closed-form expressions for several Revan
degree-based topological indices of the supramolecular Fuchsine sheet, including the first
and second Revan indices(𝑅1 and 𝑅2), Atomic Bond Connectivity Revan index(ABCR),
Geometric-Arithmetic Revan index(𝐺𝐴𝑅), the first and second hyper Revan indices(𝐻𝑅1
and 𝐻𝑅2), the first and second modified Revan indices(𝑚𝑅1 and 𝑚𝑅2) , forgotten Revan
index(𝐹𝑅). A detailed numerical and graphical analysis demonstrates that all these indices
increase monotonically with the parameters m and n, reflecting the scaling behaviour of the
supramolecular structure. Among the indices studied, the first hyper Reven index exhibits
the highest values, whereas the first modified Revan index yields the lowest. Our findings
provide a comprehensive mathematical characterization of the supramolecular Fuchsine
graph, offering valuable insights for modelling and predicting the properties of complex
chemical systems using topological descriptors.
