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A variation of zero-divisor graphs

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Gupta R., Sen M., Ghosh S.

Discussiones Mathematicae - General Algebra and Applications

2015

tom 35

nr 2

159-176

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.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Ghosh S.

1 Gupta R.

1 Sen M.

1 2015

Wyniki wyszukiwania

A variation of zero-divisor graphs