New Graph Descriptors Predict Graph Energy Across Graphene Allotropes

A pair of mathematicians at Vellore Institute of Technology in Chennai has reported a refined way to characterize three non-hexagonal graphene allotropes using graph-theoretical descriptors, entropy measures, and regression models that predict graph energy with high accuracy. The study, published in Frontiers in Chemistry, compares conventional degree-based topological indices against a modified reverse degree framework for three carbon frameworks known as PHOTH-graphene, sulphur-doped graphene, and pentahexoctite-graphene.

Graphene itself is a one-atom-thick sheet of carbon atoms in a hexagonal lattice, with carbon-carbon bonds of roughly 1.42 angstroms, exceptional mechanical strength, and excellent electrical and thermal conductivity. Its zero band gap and limited chemical reactivity, however, have motivated researchers to design alternative graphene-based architectures. The three structures examined here break away from purely six-membered rings: PHOTH-graphene fuses four-, five-, six-, seven-, and eight-membered rings in an orthorhombic lattice with ten atoms per primitive cell; sulphur-doped graphene rests on a rectangular lattice with eight carbon atoms per unit cell and two Dirac cones in its band structure; and pentahexoctite-graphene arranges five-, six-, and eight-membered rings in a fully planar configuration whose phonon spectrum shows no imaginary frequencies, confirming dynamic stability.

Topological descriptors are numerical summaries of molecular connectivity derived purely from the graph of a structure, where atoms become vertices and bonds become edges. Such indices underpin quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) analysis, linking molecular architecture to material behavior. The authors employed a modified reverse degree-based approach, introducing a variable reversing parameter that alters how the degree sequence of the graph is transformed, and computed a family of indices including modified Zagreb, forgotten, geometric-arithmetic, bi-Zagreb, tri-Zagreb, and harmonic-geometric variants for graphene sheets of varying dimensions.

The computations relied on partitioning the edges of each graphene graph into equivalence classes according to the degrees of their endpoint vertices. All three frameworks have a maximum vertex degree of three, so the edges fall into classes labeled (2,2), (2,3), and (3,3). When the reversing parameter is applied, these classes convert in a defined way, for example (2,2) becoming (3,3). The team tabulated descriptor values for reversing parameters of one, two, and three across sheet dimensions from single-unit cells up to three-by-three arrays. Among the frameworks, PHOTH-graphene generally showed the highest descriptor values, which the authors interpret as reflecting strong structural coherence, while a subset of the modified reverse degree descriptors consistently produced lower values that may simplify calculations for very large sheets.

To test whether the new descriptors actually distinguish one structure from another, the researchers performed a sensitivity analysis. This matters because a recurring weakness of topological indices is that distinct molecules can share identical descriptor values, blunting their usefulness. The sensitivity measure, defined as the fraction of considered structures uniquely characterized by a descriptor, exceeded 0.92 for every configuration examined and climbed above 0.99 for larger sheets, indicating strong discriminating power that improves as the frameworks grow.

The second pillar of the study is entropy. Building on Shannon’s information entropy, which quantifies uncertainty in a probability distribution, the authors computed graph entropy from the edge probabilities implied by their modified reverse degree descriptors. Because raw entropy values are complicated by differences in the number of edges between frameworks, they introduced a scaled bond-wise entropy that normalizes by edge count. On this per-bond basis, PHOTH-graphene exhibited the highest normalized entropy, consistent with its more intricate ring architecture, while sulphur-doped and pentahexoctite graphene showed lower values indicative of simpler structures. Notably, as sheet dimensions increased, the normalized entropies of all three frameworks gradually converged, suggesting that large-scale versions of these allotropes approach similar structural complexity regardless of their local ring patterns.

The third component addresses graph energy, a spectral invariant defined as the sum of the absolute eigenvalues of a graph’s adjacency matrix. Graph energy is associated with the total pi-electronic energy within Hückel molecular orbital theory, making it relevant to electronic properties and chemical reactivity. Computing adjacency matrices for very large sheets with software such as newGRAPH becomes cumbersome, so the authors built regression models that predict graph energy directly from descriptor values. Combining all three frameworks into a single dataset, they fitted linear and quadratic models and evaluated them with correlation coefficients, adjusted R-squared, and error metrics including mean absolute error, mean squared error, root mean squared error, and mean absolute percentage error.

The modified reverse bi-Zagreb geometric index emerged as the most effective predictor, achieving the highest correlation coefficient and lowest standard error. For the fitted models, correlation coefficients reached 0.999 or higher, with mean absolute percentage errors as low as roughly 0.45 percent for PHOTH-graphene and 0.52 percent for sulphur-doped graphene under the reversing parameter of two. Predicted energies for higher-dimensional sheets, checked against values computed with newGRAPH, tracked the actual figures closely, and leave-one-out cross-validation, in which each data point is excluded in turn and the model rebuilt from the remainder, confirmed the models’ stability with high cross-validated correlation and low root mean square error of cross-validation.

The team also analyzed spectral properties, including pi-electron energy per bond, delocalization energy per bond, and spectral diameter, computed with newGRAPH and MATLAB. All three quantities increased with sheet size for every framework, indicating greater pi-electron delocalization, resonance stabilization, and a wider spread of eigenvalues as the structures grow. The authors caution that their findings concern mathematical models of idealized graphene frameworks rather than measured properties of synthesized materials, and the entropy comparisons are normalized constructs rather than thermodynamic measurements. Still, they argue that combining graph-theoretical descriptors with statistical modeling offers a computationally efficient route to screening novel carbon allotropes, and potentially a foundation for machine-learning applications in computational chemistry and materials design.

Subject of Research: Graph-theoretical topological descriptors, entropy measures, and regression models for predicting graph energy in non-hexagonal graphene allotropes

Article Title: Comparison of degree and modified reverse degree descriptors incorporating entropy and graph energy analysis for graphene allotropes

Article References: Celine, M. R., & Greeni, A. B. (2026). Comparison of degree and modified reverse degree descriptors incorporating entropy and graph energy analysis for graphene allotropes. Frontiers in Chemistry, 14, Article 1835238. https://doi.org/10.3389/fchem.2026.1835238

Image Credits: AI Generated

DOI: 10.3389/fchem.2026.1835238

Keywords: graphene allotropes, topological descriptors, graph energy, Shannon entropy, QSPR modeling, regression analysis, PHOTH-graphene, sulphur-doped graphene, pentahexoctite-graphene, spectral properties, cross-validation, mathematical chemistry