For a commutative semigroup S with 0, the zero-divisor graph of S denoted by Γ(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study the case where the graph Γ(S) is complete r-partite for a positive integer r. Also we study the commutative semigroups which are finitely colorable.

Let L be a lattice with the least element 0. An element x ∈ L is a zero divisor if x∧ y = 0 for some y ∈ L∗ = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph Γ(L) to L with vertex set Z(L)∗ = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y = 0. In this paper, we obtain certain properties and dia...

For a commutative semigroup S with 0, the zero-divisor graph of S denoted by Γ(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study median and center of this graph. Also we show that if Ass(S) has more than two elements, then the girth of Γ(S) is three.

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in   R     W R  , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and   W R  a bR  b aR  . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs    ...