In 1988, Beck [10] introduced the notion of coloring of a commutative ring R. Let G be a simple graph whose vertices are the elements of R and two vertices x and y are adjacent if xy = 0. The graph G is known as the zero divisor graph of... more
In 1988, Beck [10] introduced the notion of coloring of a commutative ring R. Let G be a simple graph whose vertices are the elements of R and two vertices x and y are adjacent if xy = 0. The graph G is known as the zero divisor graph of R. He conjectured that, the chromatic number χ(G) of G is same as the clique number ω(G) of G. In 1993, Anderson and Naseer [1] gave an example of a commutative local ring R with 32 elements for which χ(G) > ω(G). Further, this concept of zero divisor graphs is well studied in algebraic structures such as rings, semigroups; see Anderson et. al. [1, 2], F. DeMeyer et. al. [14, 15], LaGrange [31, 32], Redmond [53, 54], and in ordered structure such as lattices, meet-semilattices, posets and qosets; see Alizadeh et. al. [9], Estaji and Khashyarmanesh [17], Halas and Langer [21], Joshi et. al. [27, 28, 29], Lu and Wu [37], Nimbhorkar et. al. [48, 49, 68]. In this Thesis, we deal with the basic properties such as connectivity, diameter, girth (gr), ec...
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In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the... more
In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.
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In this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S\Z(S) = {1}, then Γ(B) ≌ Γ(S) if and only if B ≌ S. Further, we characterize the graphs which can be realized as zero divisor... more
In this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S\Z(S) = {1}, then Γ(B) ≌ Γ(S) if and only if B ≌ S. Further, we characterize the graphs which can be realized as zero divisor graphs of Boolean posets.
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In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its... more
In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.