Clique and distance analysis in the undirected power graph of Zn

Abstract

In this paper, we investigate the structural properties of the undirected power graph (GZn) associated with the finite cyclic group Zn. We begin by establishing the necessary and sufficient conditions under which (GZn) It is bipartite and characterizes its clique structure. Explicit formulas are derived for the degree of a vertex based on the subgroup structure of Zn, leading to a criterion for identifying the minimum degree in the graph. We also analyze distance-related parameters, including radius, diameter, and average distance, and explore conditions under which these attain extremal values. The number of vertex pairs at distance two is computed for specific group orders, such as (GZpq) and (GZp2q), revealing intricate combinatorial relationships. These results enhance the understanding of how algebraic properties of Zn influence the topology of its power graph.