Degree and Order of the Partitioned Zero Divisor Graphs of Five Radical Zero Commutative Completely Primary Finite Rings of Characteristic p ^{n Ndago F. Omondi, Hezron S. Were}

Let R be a commutative completely primary finite ring with an identity 1 ≠ 0 and a unique maximal ideal Z ( R ) such that ( Z ( R )) ⁵ = (0) and ( Z ( R )) ⁴ ≠ (0). Let Γ( R ) denote the zero‐divisor graph whose vertex set V (Γ( R )) = Z ( R ) ^∗ consists of all nonzero zero‐divisors of R , where two distinct vertices u , v ∈ V (Γ( R )) are adjacent if and only if u v = 0. In this paper, we establish a novel partition of Z ( R ) ^∗ into mutually disjoint subsets based on the adjacency properties and algebraic structure of the ring R . We determine the exact order of these subsets and compute the degree of every vertex within these partitions for characteristics Char ( R ) = p ⁿ (for n = 1, 2, 3). Illustrative examples and graph representations supporting the structural findings are also presented.