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On the intersection graphs of ideals of direct product of rings

100%

Rad N., Jafari S., Ghosh S.

Discussiones Mathematicae - General Algebra and Applications

2014

tom 34

nr 2

191-201

In this paper we first calculate the number of vertices and edges of the intersection graph of ideals of direct product of rings and fields. Then we study Eulerianity and Hamiltonicity in the intersection graph of ideals of direct product of commutative rings.

A variation of zero-divisor graphs

100%

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

2 Discussiones Mathematicae - General Algebra and Applications

2 Ghosh S.

1 Gupta R.

1 Jafari S.

1 Rad N.

1 Sen M.

1 2015

1 2014

Wyniki wyszukiwania

On the intersection graphs of ideals of direct product of rings

A variation of zero-divisor graphs