Line graphs associated to the maximal graph

Let R be a commutative ring with identity. Let G(R) denote the maximal graph associated to R, i.e., G(R) is a graph with vertices as the elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. Let Γ(R) denote the restriction of G(R) to non-unit elements of R. In this paper we study the various graphical properties of the line graph associated to Γ(R), denoted by (Γ(R)) such that diameter, completeness, and Eulerian property. A complete characterization of rings is given for which diam(L(Γ(R)))=diam(Γ(R)) or diam(L(Γ(R)))diam(Γ(R)). We have shown that the complement of the maximal graph G(R), i.e., the comaximal graph is a Euler graph if and only if R
has odd cardinality. We also discuss the Eulerian property of the line graph associated to the comaximal graph.