The total graph and regular graph of a commutative ring

Let RR be a commutative ring. The total graph   of RR, denoted by T(Γ(R))T(Γ(R)) is a graph with all elements of RR as vertices, and two distinct vertices x,y∈Rx,y∈R, are adjacent if and only if x+y∈Z(R)x+y∈Z(R), where Z(R)Z(R) denotes the set of zero-divisors of RR. Let regular graph   of RR, Reg(Γ(R))Reg(Γ(R)), be the induced subgraph of T(Γ(R))T(Γ(R)) on the regular elements of RR. Let RR be a commutative Noetherian ring and Z(R)Z(R) is not an ideal. In this paper we show that if T(Γ(R))T(Γ(R)) is a connected graph, then diam(Reg(Γ(R)))⩽diam(T(Γ(R))). Also, we prove that if RR is a finite ring, then T(Γ(R))T(Γ(R)) is a Hamiltonian graph. Finally, we show that if SS is a commutative Noetherian ring and Reg(S)Reg(S) is finite, then SS is finite.