Degree and Order of the Partitioned Zero Divisor Graphs of Five Radical Zero Commutative Completely Primary Finite Rings of Characteristic pn, 1≤n≤3

Let R be a commutative completely primary finite ring with an identity 1≠0 and a unique maximal ideal ZR such that ZR5=0 and ZR4≠0. Let ΓR denote the zero-divisor graph whose vertex set VΓR=ZR∗ consists of all nonzero zero-divisors of R, where two distinct vertices u,v∈VΓR are adjacent if and only if uv=0. In this paper, we establish a novel partition of ZR∗ 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=pn (for n=1,2,3). Illustrative examples and graph representations supporting the structural findings are also presented.