Geodesic
A variation of zero-divisor graphs
Discussiones Mathematicae. General Algebra and Applications, Tome 35 (2015) no. 2, pp. 159-176

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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 ℤ₂^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., P_n), or cycles (i.e., 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.
Keywords: rings, zero-divisor graphs, finite fields 
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Gupta, Raibatak; Sen, M.; Ghosh, Shamik. A variation of zero-divisor graphs. Discussiones Mathematicae. General Algebra and Applications, Tome 35 (2015) no. 2, pp. 159-176. http://geodesic.mathdoc.fr/item/DMGAA_2015_35_2_a3/