Let R be a commutative ring and let Z (R) denotes the set of zero-divisors of R. The triple total graph of R, denoted by TT (R), is a simple graph whose vertex set is R∖0. Two distinct vertices v1 and v2 are adjacent in TT (R) if and only if v1+v2∉Z (R) and there exists v3∈R∖0, with v3≠v1 and v3≠v2, such that v1+v3∉Z (R), v2+v3∉Z (R), and v1+v2+v3∈Z (R). In this paper, we investigate the structural properties of the graph TT (Zn). We show that if n is even with n 2, then TT (Zn) is an empty graph. When n is prime with 2 n 11, the graph TT (Zn) is disconnected. In contrast, for prime integers n≥ 11, the graph becomes connected with diam (TT (Zn) ) =2 and gr (TT (Zn) ) =3. Moreover, each vertex has degree n−5, implying that the graph is (n−5) -regular and consequently both Eulerian and Hamiltonian. These results illustrate how the arithmetic nature of n determines the global structure of the triple total graph.
Kurniawan et al. (Fri,) studied this question.