Numerical Invariants Of Nilpotent Graphs In Integer Modulo Rings

Graph theory offers a robust framework for examining algebraic structures, especially rings and their elements. This paper focuses on the nilpotent graph of rings of the form Z_pk​, where p is a prime and k∈N, investigating both their structural and numerical properties. We begin by characterizing the nilpotent elements in these rings and examining their relationship to ring ideals. The study then presents theoretical results on key graph invariants, including connectivity, chromatic number, clique number, and specific subgraph configurations. To complement these, we also analyze numerical invariants such as edge count and degree distribution, which reveal deeper connections between ring-theoretic and graph-theoretic properties. Our results highlight consistent structural patterns in nilpotent graphs of Z_pk ​and provide a concrete contribution to algebraic graph theory by bridging properties of commutative rings and their associated graphs.