The Girth of the Total Graph of ℤ n

Abstract

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