Topological indices describe mathematical invariants of molecules in mathematical chemistry. M-polynomials of chemical graph theory have freedom about the nature of molecular graphs and they play a role as another topological invariant. Social networks can be both cyclic and acyclic in nature. We develop a novel application of M-polynomials, the ( m , n , r ) -agent recruitment graph where n > 1 , to study the relationship between the Dunbar graphs of social networks and the small-world phenomenon. We show that the small-world effects are only possible if everyone uses the full range of their network when selecting steps in the small-world chain. Topological indices may provide valuable insights into the structure and dynamics of social network graphs because they incorporate an important element of the dynamical transitivity of such graphs.

中文翻译：

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社交网络中邓巴图的M多项式

拓扑指数描述了数学化学中分子的数学不变量。化学图论的 M 多项式对分子图的性质具有自由度，并且它们扮演着另一个拓扑不变量的角色。社交网络本质上既可以是循环的，也可以是非循环的。我们开发了 M 多项式的新应用，即 ( m , n , r ) 代理招募图，其中 n > 1，以研究社交网络的邓巴图与小世界现象之间的关系。我们表明，只有当每个人在选择小世界链中的步骤时都使用其网络的全部范围时，小世界效应才有可能。拓扑索引可以为社交网络图的结构和动态提供有价值的见解，因为它们包含了此类图的动态传递性的重要元素。