The Girth of the Total Graph of ℤ _n

Abstract

Let be a commutative ring with a non-zero identity, and ( ) is a set of zero-divisors of . The total graph of , denoted Γ ( ), is an (undirected) graph with all elements as vertices of Γ ( ) and for distinct vertices , ∈ are adjacent if and only if + ∈ ( ). The girth of Γ ( ) is the length of the shortest cycle in Γ ( ), its denoted by ( Γ ( )). In this paper, we discuss the characterization of the total graph of ℤ , Γ (ℤ )and ( Γ (ℤ )).