Non-Nilpotent Graph of 2×2 Upper Triangular Matrix Rings over Z4p for Prime Numbers p

Suppose R represents an arbitrary ring. Let Nil(R) be the collection of all nilpotent elements in R. The non-nilpotent graph, symbolized by GNN(R), is defined over the vertex set R Nil(R), where two distinct vertices form an edge whenever their sum belongs to Nil(R). Furthermore, let T2(Zn) denote the ring of 2 2 upper triangular matrices with entries in Zn. This paper investigates the structure of GNN(T2(Z4p)), where p 3 is a prime number. The main findings reveal that GNN(T2(Z4p)) decomposes into a disjoint union of complete subgraphs and complete bipartite subgraphs. Additionally, key structural invariants of this graph, namely its clique number, girth, diameter, and domination number, are analyzed, along with an examination of its Eulerian and Hamiltonian properties.