Tensor Product Of Zero-divisor Graphs With Finite Free Semilattices

Γ(S LX) is defined and has been investigated in (Toker, 2016). In this paper our main aim is to extend this study over Γ(S LX) to the tensor product. The diameter, radius, girth, domination number, independence number, clique number, chromatic number and chromatic index of Γ(S LX1 ) ⊗ Γ(S LX2 ) has been established. Moreover, we have determined when Γ(S LX1 ) ⊗ Γ(S LX2 ) is a perfect graph.


Introduction
Let G be a graph then edge set of G denoted by E(G) and vertex set of G denoted by V(G).Let G 1 and G 2 be graphs, tensor product of G 1 and G 2 has vertex set V(G 1 ) × V(G 2 ) and has edge set {(u 1 , v 1 )(u 2 , v 2 ) : u 1 u 2 ∈ E(G 1 ) and v 1 v 2 ∈ E(G 2 )}, and it is denoted by G 1 ⊗ G 2 .Let G 1 and G 2 be connected graphs then G 1 ⊗ G 2 is connected if and only if either G 1 or G 2 contains an odd cycle (Weichsel, 1962).Also it is clear that Firstly zero-divisor graph on a commutative semigroup S with 0 was studied by Demeyer and his friends (DeMeyer et all., 2002;DeMeyer et all, 2005).Let the set of zero divisor elements in S is Z(S ), the zero-divisor graph Γ(S ) is defined as an undirected graph with vertices Z(S ) \ {0} and the vertices x and y are adjacent with a single edge if and only if xy = 0. Always Γ(S ) is a connected graph (DeMeyer et all., 2002).
Let X be a finite non-empty set.The free semilattice on a set X is the finite powerset of X except the empty set and operation is union of sets.We show it with S L X .Then S L X is a commutative semigroup of idempotents with the multiplication A • B = A ∪ B for all A, B ∈ S L X .In zero-divisor graph of S L X , any two distinct vertices A and B are adjacent with the rule A ∪ B = X.In a recent study, Γ(S L X ) has been investigated in (Toker, 2016).
We know that if |X| ≥ 3 then Γ(S L X ) contains an odd cycle (Toker, 2016).Let X 1 and X 2 be non-empty and finite sets and let Γ(S L X 1 ) be zero-divisor graph associated to S L X 1 and Γ(S L X 2 ) be zero-divisor graph associated to S L X 2 .In this paper, without loss of generality we assume that ) is connected graph and in this paper we have researched girth, diameter, radius, dominating number, clique number, chromatic number, chromatic index, independence number and perfectness of this graph.

Some Properties of Γ(S L X
Let G be a simple graph, the distance (length of the shortest path) between two vertices u, v in G is denoted by d G (u, v).In a connected simple graph the maximum distance (lenght of the shortest path) between v and any other vertex u in G is eccentricity of a vertex v,it is denoted by ecc(v), so that is and it is denioted by diam(G).Moreover radius of G is defined by and it is denoted by rad(G).The girth of a graph is the length of a shortest cycle contained in the graph, and it is denoted by gr(G).If there is not any cycle in a graph, then its girth is defined to be infinity.
The degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice, degree of vertex v ∈ V(G) is denoted by deg G (v).Among all degrees, the maximum degree is denoted by ∆(G) and the minimum degree is denoted by δ(G).In a graph, the vertex of maximum degree is called delta-vertex and the set of delta-vertices of G denoted by Λ G .In a graph, an independent set or stable set is a set of vertices in a graph, no two of which are adjacent.Independence number of G is denoted by α(G) and it is defined by In a graph, a dominating set for a graph G is a subset D of V(G) such that every vertex not in D is adjacent to at least one member of D. The domination number of G is the number of vertices in a smallest dominating set for G, and it is denoted by γ(G), so dominating number of G is The open neighbourhood of a vertex v ∈ V(G) is the set of vertices which are adjacent to v and it is denoted by In this section we mainly deal with some graph properties of Γ(S L X 1 )⊗Γ(S L X 2 ) namely diameter, radius, girth, domination number and independence number.
We use the notation A = (X i \ A) for all A ⊆ X i (i = 1, 2), and we use the notation Moreover for convenience we use , in other cases we have same results with i).This case we take 2−partition of B 1 , we say M and N. We have a path (A 1 , B 1 ) (A 1 , B 1 ) has only one adjacent and it is (A 2 , B 1 ) and (A 2 , B 1 ) has only one adjacent and it is (A 1 , B 1 ) and they are different vertices and they are not adjacent, moreover d((A 2 , B 1 ), (A 1 , B 1 )

2
, and C is an independence set for Let G be a graph.Clique is the each of the maximal complete subgraphs of G.The number of all the vertices in any clique of G is clique number and it is denoted by ω(G).The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color and it is denoted by χ(G).It is well-known that for any graph G (Chartrand & Zhang, 2009).
The core of a graph G is defined to be the largest induced subgraph of G such that each edge in core is part of a cycle and it is denoted by G ∆ .Finally, let M be a subset of E(G) for a graph G, if there is no two edges in M which are adjacent then M is called a matching.
for 1 ≤ i j ≤ n.For each 1 ≤ k ≤ n if we choose a different colour for each Q k and assign the chosen colour to the all |B| ≥ 2 so there exists 2−partition of A, we say A 1 and A 2 and there exists 2−partition of B, we say B 1 andB 2 .Thus (A, B)−(A∪A 1 , B∪B 1 )−(A∪A 2 , B∪B 2 )−(A, B) is a cycle.Let |A| = 1, |B| ≥ 2 then there exists ∅ C A so we have a cycle (A, B)−(A, B∪B 1 )−(A∪C, B)−(A, B∪B 2 )−(A, B).If |A| ≥ 2, |B| = 1,we can find a cycle similar way.Moreover Γ 1 ⊗Γ 2 is simple graph and from its definition gr(Γ 1 ⊗Γ 2 ) of those edges whose endpoints are in V ′ .For each induced subgraphH of G, if χ(H) = ω(H), then G is called a perfect graph.The complement or inverse of a graph G is a graph on the same vertices such that two distinct vertices are adjacent if and only if they are not adjacent in G, the complement of G is denoted by G c .A graph G is called Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one.The edges are called adjacent if they share a common end vertex.An edge coloring of a graph is an assignment of colors to the edges of the graph so that no two adjacent edges have the same color.The minimum required number of colours for and edge colouring of G is called the chromatic index of G and it is denoted by χ ′ (G).Vizing gave a fundamental theorem for that, for any graph G, we have ∆