Spectral bounds for Sombor and Sombor energy indices: A graph-theoretic study of neurotransmitter molecular networks

This paper applies the recently introduced Sombor index and its spectral extension, the Sombor energy, to model and analyze the structural complexity of neurotransmitter molecular graphs. Let $G$ denote a molecular graph whose vertices and edges correspond to atoms and covalent bonds, respectively. For each $G$, we compute $SO\left(G\right)$ and $SOE\left(G\right)$, and derive degree-based, spectral-radius, and Frobenius-norm bounds to quantify molecular irregularity. Unlike traditional indices such as Zagreb or Wiener, Sombor descriptors incorporate both degree heterogeneity and geometric weighting, offering refined sensitivity to branching and aromaticity. Comparative analysis across inhibitory (glycine, GABA) and excitatory or modulatory (dopamine, serotonin, norepinephrine) neurotransmitters reveals that higher Sombor measures correspond to greater structural and functional complexity. These results confirm that Sombor-based descriptors capture biologically interpretable differences in molecular organization. The study thereby extends spectral graph theory to neurochemical systems, providing a quantitative framework for cheminformatics, drug design, and functional classification of neurotransmitters.