Clar's goblet frameworks: network topologies, entropies, graph energies, and the existence of isentropic frameworks

The topology of $\pi$ -conjugated carbon nanostructures strongly influences their electronic, magnetic, and spectral properties, thereby enabling applications in molecular electronics and spintronics. Graphene nanoribbons (GNRs), which are quasi-one-dimensional carbon systems, exhibit tunable electronic behavior. In GNRs, Clar's goblet (CG) represents a non-Kekule open shell $\pi$ -electron framework with a triplet ground state. These systems exhibit a zero HOMO-LUMO gap within the Hückel approximation, and thus, they exhibit a low-lying excited singlet state. Their open-shell structures with facile $\pi$ -conjugated networks lend excitonic features to the network making them ideal candidates for spintronics. In this study, we derive generalized expressions for distance and degree-based descriptors for their network topology using double-weighted graph-theoretical techniques. In addition, Shannon's entropies and graph energies are evaluated to contrast their relative order and energetics. Our computations demonstrate the existence of isentropic frameworks. Furthermore, quantitative structure-property relationships are determined for obtaining spectral graph energies through the topological indices, offering a technique to predict the graph spectral energies of larger networks without explicit computations of their graph spectra.