The total graph and regular graph of a commutative ring

Abstract

Let R be a commutative ring. The total graph of R , denoted by T ( Γ ( R ) ) is a graph with all elements of R as vertices, and two distinct vertices x , y ∈ R , are adjacent if and only if x + y ∈ Z ( R ) , where Z ( R ) denotes the set of zero-divisors of R . Let regular graph of R , R e g ( Γ ( R ) ) , be the induced subgraph of T ( Γ ( 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 ( Γ ( R ) ) is a connected graph, then diam ( R e g ( Γ ( R ) ) ) ⩽ diam ( T ( Γ ( R ) ) ) . Also, we prove that if R is a finite ring, then T ( Γ ( R ) ) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and R e g ( S ) is finite, then S is finite.