Let R be a commutative ring. The total graph of R, denoted by T .???? .R// is a graph with all elements of R as vertices, and two distinct vertices x; y 2 R, are adjacent if and only if x C y 2 Z.R/, where Z.R/ denotes the set of zero-divisors of R. Let regular graph of R, Reg.???? .R//, be the induced subgraph of T .???? .R// on the regular elements of R. Let R be a commutative Noetherian ring and Z.R/ is not an ideal. In this paper we show that if T .???? .R// is a connected graph, then diam.Reg.???? .R/// 6 diam.T .???? .R///. Also, we prove that if R is a finite ring, then T .???? .R// is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg.S/ is finite, then S is finite.
