Sigma Bond Calculation via Euler’s Formulafor Planar and Polyhedral Molecular Graphs

Abstract Sigma (σ) bonds form the fundamental backbone of molecular connectivity, repre-senting the first covalent bond between two atoms in single, double, or triple bonds.Determining the number of σ bonds in a molecule is a basic yet essential task in struc-tural chemistry, with implications for stability, reactivity, and computational modeling.Traditional approaches require either manual bond counting from chemical drawingsor computational analysis of 3D structures, both of which can be time-consuming andprone to error for large, complex, or unconventional molecules.In this work, we present a mathematically rigorous and computationally efficientformula that predicts the number of σ bonds from three simple topological parameters:the total number of atoms (T ), the number of rings or faces (R), and a polyhedral indi-cator (ϕ). Derived directly from Euler’s characteristic for molecular graphs, the formula: σ = T +R −ϕ Achieves perfect agreement with known values across a diverse validationset of 50 molecules, including simple organics, heterocycles, ions, biomolecules, poly-hedral clusters, and extended frameworks. We provide a complete proof, schematicillustrations for planar and spherical embeddings, and an algorithmic flowchart. Thesimplicity, accuracy, and broad applicability make it a powerful tool for chemical in-formatics, education, and rapid molecular screening. For connected genus-0 moleculargraphs. Limitations for higher-genus/disconnected systems are noted.