Some new results on the graph Γ1(R) over a ring R

In this paper we continue our study of the graph Γ1(R)Γ1(R) defined over a ring R   with unity. Γ1(R)Γ1(R) is a simple undirected graph having all the non-zero elements of R as its vertices and two vertices a, b   are adjacent if and only if either ab=0ab=0 or ba=0ba=0 or a+ba+b is a unit in R  . Concentrating on the graph Γ1(Zn)Γ1(Zn), we look at several properties like degrees, girth, Eulerianity, planarity etc. We also look at some properties of the graph Γ1(R)Γ1(R) taken over matrix rings and of the graph Γ1(F)Γ1(F) over a finite field F.