| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-agent recruitment graph where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, 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.
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