Mathematical graph tools reveal electronic secrets of butterfly-shaped nanographenes

Graphene has spent two decades dazzling scientists with its extraordinary blend of thermal, optical, mechanical and electronic properties, and its high electron mobility continues to make it a star candidate for next-generation field-effect transistors. But as researchers learn to sculpt graphene into ever more elaborate shapes at the molecular scale, a pressing question has emerged: how can we predict the electronic behaviour of these intricate structures without running prohibitively expensive quantum chemical calculations every time? A new study published in Results in Chemistry offers an elegant answer, deploying the machinery of graph theory and information theory to dissect a family of molecules that look, quite literally, like butterflies.

The structures in question are nanographene butterfly structures, open-shell polycyclic aromatic hydrocarbons whose carbon skeletons spread into two wing-like lobes. These molecules belong to a class of materials that has seen remarkable recent progress in design strategies and synthetic methodologies. In a striking experimental milestone cited by the authors, researchers combined electron–electron interactions with topological frustration to create the largest completely fused open-shell nanographene to date: a butterfly-shaped tetraradical assembled on a gold surface. Structures of this kind are semi-one-dimensional, extended monolayer segments of graphene with a high length-to-width ratio, closely related to graphene nanoribbons, sp2-hybridized honeycomb lattices whose edge configuration plays a crucial role in determining their electronic and structural properties.

To tame the complexity of these molecules, the research team, led by Simili Abraham and colleagues including Jyothish K, Tony Augustine, Bibin K Jose, Fikadu Tesgera Tolasa and Aseel Smerat, abstracted each butterfly molecule into a mathematical graph. Atoms become vertices and chemical bonds become edges. A nanographene butterfly graph of dimension ω, where ω is greater than one, contains precisely 10ω² + 8ω + 4 vertices and 15ω² + 7ω + 4 edges. This clean combinatorial description allowed the team to partition every edge of the graph according to the degrees of its two endpoint vertices, a classification that turns out to be remarkably simple: only three edge types exist, connecting vertices of degree 2–2, 2–3 and 3–3, with frequencies that can be written as exact formulas in ω.

From this partition the authors derived closed-form expressions for fourteen degree-based topological indices, single numbers that condense structural information about a molecule into a computationally cheap descriptor. The roster includes some of the most celebrated indices in chemical graph theory: the Randić and reciprocal Randić indices, the first and second Zagreb indices and their reduced and hyper variants, the augmented Zagreb index, the atom-bond connectivity index, the harmonic and sum-connectivity indices, the Sombor index introduced by Gutman in 2021, the geometric-arithmetic index and the inverse sum index. Because these descriptors depend only on vertex degrees rather than full quantum mechanical wavefunctions, they are especially suitable for large-scale molecular screening, avoiding the need for time- and resource-intensive computations.

The numerical behaviour of these indices as the butterfly grows from dimension two to dimension eight reveals a clear hierarchy. The hyper-Zagreb index is the most explosive, climbing quadratically from 2,304 at ω = 2 to nearly 35,000 at ω = 8, far outpacing every other descriptor and showing acute sensitivity to the rising vertex degrees that accompany growth. The first Zagreb, second Zagreb, augmented Zagreb and Sombor indices follow with substantial increases, while the reciprocal Randić, reduced reciprocal Randić, reduced second Zagreb and inverse sum indices grow more modestly. This sensitivity ranking tells chemists which descriptors are most responsive to structural size, a practical guide for anyone using topological indices to model how molecular properties scale.

The study’s second mathematical pillar is entropy, borrowed from Claude Shannon’s foundational 1948 theory of communication. Shannon entropy quantifies the information carried by a probability distribution, and here the distribution in question is built from the contributions of individual edges to a chosen topological index. For each of the fourteen degree-based indices, the authors constructed a probability function over the edges of the butterfly graph and computed the associated graph entropy, producing a family of information-theoretic measures of structural complexity. The resulting entropy expressions, derived as exact theorems for the butterfly family, capture how evenly or unevenly the structural contributions are spread across the molecular framework.

A subtle and somewhat counterintuitive pattern emerged from the entropy analysis. Although the number of vertices and edges rises steeply with dimension, the entropy values increase only slowly and predictably, with the bars in comparative plots remaining nearly parallel across all entropy types. The reason lies in the regularity of the degree distribution: as the butterfly expands, the pattern of vertex degrees stays fairly consistent, a characteristic typical of structured nanomaterials such as nanographene. Even as the graphs grow larger and more complex in absolute terms, their informational complexity, as measured by Shannon entropy over degree contributions, evolves in an orderly, almost serene manner.

The third and most chemically consequential strand of the work concerns π-electron energy, a spectral descriptor that characterizes the electronic structure and degree of π-electron delocalization in conjugated molecular systems. In the Hückel framework, the total π-electron energy of a molecular graph equals the sum of the absolute values of the eigenvalues of its adjacency matrix, expressed in units of the resonance integral β. The authors generated butterfly structures for dimensions up to eight, built the corresponding adjacency matrices, and computed their spectra. Crucially, they also evaluated the size-normalized quantity Eπ divided by n, the number of vertices, which represents π-electron energy per bond and is the fairer comparator between molecules of different sizes.

The normalized values tell a compelling story of progressive electronic enrichment. Rising systematically from 1.45903 at dimension two to 1.53209 at dimension eight, the per-bond π-electron energy indicates increasingly extended π-electron delocalization as the butterfly framework grows. The total energy, meanwhile, grows predominantly rather than linearly with ω, tracking the quadratic increase in carbon count, which is precisely why the authors argue that the raw total energy should never be used alone as a size-independent measure of stability. The normalized quantity strips away this trivial size effect and exposes the genuine energetic evolution of the conjugated framework, offering a clearer window into how these open-shell systems stabilize their delocalized electrons as they expand.

Perhaps the most forward-looking result is the marriage of the two analytical strands. When the fourteen entropy descriptors were correlated against the π-electron energies across the butterfly family, every single one produced a Pearson correlation coefficient above 0.99, an extraordinarily strong positive association. The entropy tied to the reduced second Zagreb index led the field at 0.99646, closely followed by the hyper-Zagreb, second Zagreb and reduced reciprocal Randić entropies, though the authors are careful to note that the differences are small enough that no single descriptor can yet be crowned the definitive predictor. The takeaway is profound: information-theoretic measures derived purely from the connectivity of a molecular graph carry detailed information about the energetic evolution of the π-conjugated electronic system. For a field racing to design open-shell nanographenes with tailored magnetic and electronic properties, the message is that cheap graph-theoretic descriptors, computed in moments on a laptop, can reliably track the very electronic quantities that quantum chemistry labours to deliver, opening a fast and mathematically rigorous route to screening the next generation of carbon nanostructures.

Subject of Research: Graph-theoretical and entropy-based analysis of π-electron properties in nanographene butterfly structures

Article Title: Graph theoretical and entropic insights into the π -electron properties of nanographene butterfly structures

Article References: Abraham, S., K, J., Augustine, T., Jose, B. K., Tolasa, F. T., & Smerat, A. (2026). Graph theoretical and entropic insights into the π-electron properties of nanographene butterfly structures. Results in Chemistry, 31, Article 103945. https://doi.org/10.1016/j.rechem.2026.103945

Image Credits: AI Generated

DOI: 10.1016/j.rechem.2026.103945

Keywords: nanographene, graph theory, topological indices, Shannon entropy, π-electron energy, graphene nanoribbons, chemical graph theory, polycyclic aromatic hydrocarbons, Sombor index, Zagreb indices, open-shell molecules, molecular descriptors