A variation of zero-divisor graphs

In this paper, we define a new graph for a ring with unity by extending the definition of the usual 'zero-divisor graph'. For a ring R with unity, Γ₁(R) is defined to be the simple undirected graph having all non-zero elements of R as its vertices and two distinct vertices x,y are adjacent if and only if either xy=0 or yx=0 or x+y is a unit. We consider the conditions of connectedness and show that for a finite commutative ring R with unity, Γ₁(R) is connected if and only if R is not isomorphic to ℤ₃ or ${\displaystyle \mathbb{Z}{₂}^k}$ (for any k ∈ ℕ-{1}\). Then we characterize the rings R for which Γ₁(R) realizes some well-known classes of graphs, viz., complete graphs, star graphs, paths (i.e., ${\displaystyle P_n}$), or cycles (i.e., ${\displaystyle C_n}$). We then look at different graph-theoretical properties of the graph Γ₁(F), where F is a finite field. We also find all possible Γ₁(R) graphs with at most 6 vertices.