Author

# Sima Kiani

Bio: Sima Kiani is an academic researcher from Islamic Azad University. The author has contributed to research in topics: Domination analysis & Vertex-transitive graph. The author has an hindex of 3, co-authored 4 publications receiving 21 citations.

##### Papers

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TL;DR: A classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given in this paper, where the unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit element of R.

Abstract: Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit element of R. In this paper, a classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given.

10 citations

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TL;DR: In this paper, the annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(r)\{0} and two distinct vertices I====== and J are adjacent if and only if IJ = 0.

Abstract: Let R be a commutative ring with identity and A(R) be the set of ideals with
nonzero annihilator. The annihilating-ideal graph of R is defined as the
graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I
and J are adjacent if and only if IJ = 0. In this paper, we study the
domination number of AG(R) and some connections between the domination
numbers of annihilating-ideal graphs and zero-divisor graphs are given.

10 citations

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TL;DR: In this article, the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring were investigated, where is the Ore extension of the graph?

Abstract: In this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of and , and the domination numbers of and , where is the Ore extension of , are studied.

5 citations

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TL;DR: In this article, it was shown that a unitary Cayley graph is shellable and Gorenstein if it is a Cohen-Macaulay ring, and that it is also shellable if its Stanley-Reisner ring k [ΔG(R)] is a co-occurrence.

Abstract: Let G(R) be the unitary Cayley graph corresponding to a finite commutative ring R with nonzero identity. Let ΔG(R) be the simplicial complex associated to G(R), whose faces correspond to the independent sets of G(R). We study well-coverednees of G(R) and Cohen–Macaulayness of ΔG(R), i.e., its Stanley–Reisner ring k [ΔG(R)] is a Cohen–Macaulay ring. Furthermore, we show that a unitary Cayley graph is shellable and Gorenstein if it is Cohen–Macaulay.

1 citations

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TL;DR: In this paper, the authors studied annihilator graphs of rings with equal clique number and chromatic number and gave an explicit formula for the clique-number of annihilator graph.

Abstract: Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the graph AG(R) with the vertex set Z(R)∗ = Z(R)\{0}, and two distinct vertices x and y are adjacent if and only if annR(xy)≠annR(x) ∪annR(y). In this paper, we study annihilator graphs of rings with equal clique number and chromatic number. For some classes of rings, we give an explicit formula for the clique number of annihilator graphs. Among other results, bipartite annihilator graphs of rings are characterized. Furthermore, some results on annihilator graphs with finite clique number are given.

16 citations

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TL;DR: In this paper, the essential graph of a commutative ring with identity is defined as the graph EG(R) with the vertex set Z(R ∗ = Z (R)\{0], and two distinct vertices x and y are adjacent if and only if annR(xy) is an essential ideal.

Abstract: Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The essential graph of R is defined as the graph EG(R) with the vertex set Z(R)∗ = Z(R)\{0}, and two distinct vertices x and y are adjacent if and only if annR(xy) is an essential ideal. It is proved that EG(R) is connected with diameter at most three and with girth at most four, if EG(R) contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

8 citations

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TL;DR: In this article, the metric dimension of the annihilating ideal graph of a local finite commutative principal ring and a finite principal ring with two maximal ideals has been determined, and bounds for the dimension of an annihilating-ideal graph of an arbitrary finite-convex principal ring have been derived.

Abstract: We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find the bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.
http://dx.doi.org/10.1017/S0004972712000330

6 citations

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TL;DR: In this article, the annihilating-ideal graph of R is defined to be the graph with vertex set A∗(R) √ √(R), which is the set of non-zero annihilating ideals of R.

Abstract: Let R be a commutative ring with unity 1≠0. The annihilating-ideal graph of R, denoted by 𝔸𝔾(R), is defined to be the graph with vertex set A∗(R) — the set of non-zero annihilating ideals of R and ...

4 citations