On the Total Graph and Its Complement of a Commutative Ring
Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). denotes the complement of T Γ(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi [2]. In this article, we characterize all commutative rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for to be connected whenever is connected and prove that . It is also proved that if diam(T Γ(R)) = 2, then T Γ(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al. [1].