The numerical values extracted from a graph that indicates its topology are called topological indices. A contemporary and efficient method is to compute a graph's topological indices using the graph polynomial that corresponds to it. This method of identifying degree-based topological indices involves the use of the M-polynomial. Very recently, in 2025, the hyperbolic Sombor index (HSO) was proposed and shows its chemical applicability for octane isomers and the structure sensitivity and abruptness for octane, nonane, and decane isomers, respectively. In this work, we establish the closed derivation formula for the above-mentioned index of a graph based on its M-polynomial. Additionally, we use our proposed derivation formula to calculate the hyperbolic Sombor index of a few standard graphs and chemical families. Moreover, we provide the numerical and graphical representations for the M-polynomial and the computed HSO index of the chemical families.

翻译：从图中提取的、用于表征其拓扑结构的数值称为拓扑指数。一种现代且高效的方法是利用图对应的多项式来计算其拓扑指数。这种识别基于度的拓扑指数的方法涉及使用M-多项式。最近在2025年，双曲Sombor指数（HSO）被提出，并展示了其对辛烷异构体的化学适用性，以及对辛烷、壬烷和癸烷异构体分别表现出的结构敏感性和突变性。在本工作中，我们基于图的M-多项式，建立了该指数闭式推导公式。此外，我们利用所提出的推导公式计算了几种标准图和化学族系的双曲Sombor指数。同时，我们还提供了这些化学族系的M-多项式及计算所得HSO指数的数值与图形表示。