Unveiling scale-free networks through exploring the graph theoretic properties from prime squared Jordan totient function

In this paper, we define a Jordan totient graph as an undirected graph with a vertex set [Formula: see text] consisting of first [Formula: see text] natural numbers, where [Formula: see text] is an odd prime, and an edge set [Formula: see text] consisting of a pair of distinct nodes with the property that these two along with n have no common divisor greater than one. Our investigation focuses on three primary objectives. First, we examine the applicability of standard graph-theoretic concepts to Jordan totient graphs, specifically exploring instances where the graph parameters [Formula: see text] and [Formula: see text] are derived from the Jordan totient function [Formula: see text]. Second, we develop a combinatorial formula to enumerate the number of triangles within these graphs for given [Formula: see text] and [Formula: see text]. Finally, we propose a model for scalable free social networks using Jordan totient graphs, differentiating between two groups of entities: one with internal isolation but external connections, and the other with unrestricted relationships.