Product of Semi – Lattices of Certain Graphs

Abstract

Introduction: In this article, author tries to construct a relation between graphs of product of meet-semilattices L = L_1 X L_2, where L_1 and L_2 are two semilattices and obtain some properties of such graphs. Author investigated that for meet-semilattices L_1 and L_2 has a cycle of length n-1 and n.
 Objectives: author reveals that if L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2, then it is a star graph. In this paper, we have covered some definitions, examples and theorems on zero devisor graph edge of a 4 - cycles or a 5 - cycles. Γ(L) is a star graph.
 Methods: Theorem 1.1 [9] The zero-divisor graph of a finite meet-semilattice with only one atom is the empty graph. The zero-divisor graph of the meet semilattice is the empty graph. However, this does not hold for infinite meet-semilattices with one atom. For consider, the infinite meet-semilattice, where the descending dots represent infinite descending chain. It has only one atom c but its graph Γ(L) is an infinite star graph.
 Theorem 1.2 [12] Every disconnected graph cannot be a graph of any meet- semilattice L with 0.
 Remark 1.2 A graphs of product of meet-semilattices and obtain some properties of such graphs. In this section, we consider two integral meet-semilattices L_1 and L_2 with L ∼=L_1 X L_2 and show that if ∣L_1∣ = m+1, ∣L_2∣ = n+1, then Γ(L) is the complete bipartite graph K (m, n).
 Results: If L does not contain any atom, then any edge in Γ(L) is contained in a cycle of length ≤ 6, and therefore Γ(L) is a union of 4 - cycles and 5 - cycles. Let L be a meet-semilattice with 0. If Γ(L) contains a cycle, then the core K of Γ(L) is a union of 4- cycles and 5 – cycles and any vertex in Γ(L) is either a vertex of the core K of Γ(L) or is a pendant of Γ(L). Let L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2. Then exactly one of the following holds:
 
 Γ(L) has a cycle of length n-1 or n (that is gr Γ(L)≤ n).
 Γ(L) is a star graph.
 
 Conclusions: In this article we have studied the concept of the zero-divisor graph derived from meet-semilattice L with 0 on the lines of Anderson and Livingston [6]. Also, we generalized certain results from Demeyer, Mckenzie and Schneider [18] to meet-semilattice L with 0.