Abstract   In this paper we continue our study of the graph     Γ    1    (  R  )   defined over a ring  R  with unity.     Γ    1    (  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   a  b  =  0   or   b  a  =  0   or   a  +  b   is a unit in  R . Concentrating on the graph     Γ    1    (    Z    n    )  , we look at several properties like degrees, girth, Eulerianity, planarity etc. We also look at some properties of the graph     Γ    1    (  R  )   taken over matrix rings and of the graph     Γ    1    (  F  )   over a finite field  F .
['Raibatak Sen Gupta', 'M. K. Sen']
Some new results on the graph Γ1(R) over a ring R