Zero Divisor Graphs of Posets

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], Halaš and Länger [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), eccentricity (e), radius (r), center, cut-set, clique number (ω), chromatic number (χ), domination number (γ) etc. of the