Line graph associated to total graph of idealization.

Let R be a commutative ring with identity and M be an R-module. Let Z(R) be the set of zero-divisors of R and Z(M) be the set of annihilators over M in R. The total graph of idealization, T (Γ(R(+)M)) was defined as the graph with all elements of the ring R(+)M = {(r,m) | r ∈ R,m ∈ M} as vertices where any two distinct vertices (x,m), (y, n) ∈ R(+)M are adjacent if and only if (x,m)+(y, n) ∈ Z(R(+)M), the set of zero-divisors of R(+)M. In this paper we define the line graph of total graph of idealization, denoted by L (T(Γ(R(+)M))) as the graph with all the edges of T (Γ(R(+)M)) as vertices and any two distinct vertices are adjacent if and only if their corresponding edges share a common vertex in the graph T (Γ(R(+)M)). In this paper we discuss the diameter, girth and clique number of the graph L (T(Γ(R(+)M))). We also find condition for the graph L (T(Γ(R(+)M))) to be connected when Z(R(+)M) is not an ideal of the R(+)M. [ABSTRACT FROM AUTHOR]