# The total graph of a commutative ring

##### Citations

153 citations

### Cites background from "The total graph of a commutative ri..."

...direction would be to start with a commutative ring R and use a different set of vertices or adjacency relation. For example, see [61, 75], or [ 11 ]....

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129 citations

127 citations

### Cites background from "The total graph of a commutative ri..."

...Recently Anderson and Badawi in [2] have introduced and investigated the total graph of R, denoted by T (Γ(R))....

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...In the literature, there are many papers on assigning a graph to a ring, a group, semigroup or a module (see, for example, [1–11, 14–18])....

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117 citations

### Cites background from "The total graph of a commutative ri..."

...The total graph of R denoted by T (Γ (R)) was introduced in [8], as the graph with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ Z(R)....

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...Anderson and Badawi showed that if R is a commutative ring, Z(R) is not an ideal and T (Γ (R)) is connected, then diam(Reg(Γ (R))) 6 diam(T (Γ (R))), see [8]....

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...In [8] it was proved that if R is a commutative ring such that T (Γ (R)) is connected and Z(R) is not an ideal, then diam(T (Γ (R))) 6 diam(Reg(Γ (R)))+ 2....

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...In [8] it was proved that for every commutative ring R if Z(R) is not an ideal of R, then T (Γ (R)) is connected if and only if the ideal generated by Z(R) is R (i....

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##### References

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16,023 citations

### "The total graph of a commutative ri..." refers background in this paper

...A general reference for graph theory is [8]....

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1,087 citations

### "The total graph of a commutative ri..." refers background in this paper

...In [3], Anderson and Livingston introduced the zero-divisor graph of R, denoted by Γ (R), as the (undirected) graph with vertices Z(R)∗ = Z(R)\{0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, the vertices x and y are adjacent if and only if xy = 0....

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956 citations

### "The total graph of a commutative ri..." refers background in this paper

...This concept is due to Beck [7], who let all the elements of R be vertices and was mainly interested in colorings....

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603 citations

574 citations

### "The total graph of a commutative ri..." refers background in this paper

...A standard notation for this “idealized ring” is R(+)M ; see [10] for basic properties of rings resulting from the idealization construction....

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...General references for ring theory are [10] and [11]....

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