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). Maimani et al. [ 13 Maimani , H. R. , Wickham , C. , Yassemi , S. ( 2011 ). Rings whose total graphs have genus at most one. Rocky Mountain J. Math. To appear. arXiv: 1001.5338v1[math.AC] . [Google Scholar] ] determined all isomorphism classes of finite commutative rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a commutative ring, we characterize all isomorphism classes of finite commutative rings whose total graph has genus two.

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 Anderson , D. F. , Badawi , A. ( 2008 ). The total graph of a commutative ring . J. Algebra 320 : 2706 – 2719 . [Crossref], [Web of Science ®] [Google Scholar] ]. 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 Akbari , S. , Kiani , D. , Mohammadi , F. , Moradi , S. ( 2009 ). The total graph and regular graph of a commutative ring . J. Pure Appl. Algebra 213 : 2224 – 2228 . [Crossref], [Web of Science ®] [Google Scholar] ].