THE TRIPLE IDEMPOTENT GRAPH OF THE RING Z_n - Vika Yugi Kurniawan; Bayu Purboutomo; Nughthoh Arfawi Kurdhi (Barekeng)
Let  be a commutative ring, and  denote the set of all idempotent elements of . The triple idempotent graph of , denoted by , is defined as an undirected simple graph whose vertex set . Two distinct vertices u and v in  are adjacent if and only if there exists  where  and  such that , and . This definition generalizes the notion of an idempotent divisor graph by involving a triple product, which allows deeper exploration of the combinatorial behavior of idempotents in rings. In this research, we investigate the properties of the triple idempotent graph of the ring of integers modulo n, denoted by . As a results, we establish that  and , provided that the graph is connected. Furthermore,  is Hamiltonian if n is a prime and , and Eulerian if n is a prime and .