Euler Hyperbolic Sombor Index

Abstract

In this paper, we introduce the Euler hyperbolic Sombor index and investigate its fundamental properties across various classes of graphs. We first establish general lower and upper bounds for this index and examine its relationships with several well-known topological indices, including the Euler Sombor index, the geometric-arithmetic index, the first Zagreb index, and the hyperbolic Sombor index. Furthermore, explicit formulas and extremal results are obtained for particular graph classes such as paths, cycles, stars, complete graphs, and trees, and the corresponding equality cases are characterized. In the application part, we present general combinatorial formulas for computing this index for benzenoid systems and phenylenes.