On the Domination Polynomial of Some Graph Operations

Let G be a simple graph. For any vertex V ∈ V, the open neighborhood of V is the set N(V) = {u ∈ V | uV ∈ E} and the closed neighborhood is the set N[V] = N(V) ∪ {V}. For a set S ⊆ V, the open neighborhood of S is N(S) = ⋃V∈SN(V) and the closed neighborhood of S is N[S] = N(S) ∪ S. A set S ⊆ V is a dominating set if N[S] = V, or equivalently, every vertex in V \ S is adjacent to at least one vertex in S. An i-subset of V(G) is a subset of V(G) of cardinality i. Let D(G, i) be the family of dominating sets of G which are isubsets and let d(G, i) = |D(G, i)|. The polynomialD(G, x) =


Introduction
Let  be a simple graph.For any vertex V ∈ , the open neighborhood of V is the set (V) = { ∈  | V ∈ } and the closed neighborhood is the set [V] = (V) ∪ {V}.For a set  ⊆ , the open neighborhood of  is () = ⋃ V∈ (V) and the closed neighborhood of  is [] = () ∪ .A set  ⊆  is a dominating set if [] = , or equivalently, every vertex in  \  is adjacent to at least one vertex in .An -subset of () is a subset of () of cardinality .Let D(, ) be the family of dominating sets of  which are subsets and let (, ) = |D(, )|.The polynomial (, ) = ∑ |()| =0 (, )  is defined as domination polynomial of  [1].This polynomial has been introduced by the author in his Ph.D. thesis in 2009 [2].A root of (, ) is called a domination root of .More recently, domination polynomial has found application in network reliability [3].For more information and motivation of domination polynomial and domination roots refer to [1,2].
The join  =  1 +  2 of two graphs  1 and  2 with disjoint vertex sets  1 and  2 and edge sets  1 and  2 is the graph union  1 ∪  2 together with all the edges joining  1 and  2 .The corona of two graphs  1 and  2 , is the graph  =  1 ∘  2 formed from one copy of  1 and |( 1 )| copies of  2 , where the th vertex of  1 is adjacent to every vertex in the th copy of  2 [4].the Cartesian product of two graphs  and  is denoted by ◻, is the graph with vertex set ()∪() and edges between two vertices ( 1 , V 1 ) and ( 2 , V 2 ) if and only if either In this paper, we study the domination polynomials of some graph operations.

Main Results
As is the case with other graph polynomials, such as chromatic polynomials and independence polynomials, it is natural to consider the domination polynomial of composition of two graphs.It is not hard to see that the formula for domination polynomial of join of two graphs is obtained as follows.
Theorem 1 (see [1]).Let  1 and  2 be graphs of orders  1 and  2 , respectively.Then It is obvious that this operation of graphs is commutative.Using this product, one is able to construct a connected graph  with the number of dominating sets , where  is an arbitrary odd natural number; see [5].
Let to consider the corona of two graphs.The following theorem gives us the domination polynomial of graphs of the form  ∘  1 which is the first result for domination polynomial of specific corona of two graphs.
It is easy to see that the corona operation of two graphs does not have the commutative property.The following theorem gives us the domination polynomial of  1 ∘ .Theorem 3.For every graph  of order , ( 1 ∘ , ) = (1 + )  + (, ).
Proof.In each graph of the form  1 ∘ , we have two cases for a dominating set .
Case 1.  includes  (the vertex originally in  1 ) and an arbitrary subset of the  vertices from the copy of .The generating function for the number of dominating sets of graph in this case is (1 + )  .
Case 2.  does not include  and it is exactly a dominating set of .In this case (, ) is the generating function.
The following theorem gives a formula for domination polynomial of corona products of two graphs.Theorem 4. Let  = (, ) and  = (, ) be nonempty graphs of order  and , respectively.Then Proof.By Theorem 3, it suffices to prove that ( ∘ , ) = (( 1 ∘ , ))  .In the corona of two graphs  and , every vertex  ∈  of  is adjacent to all vertices of the corresponding copy of .So, we can delete all edges in  in the corona.Therefore, the arising graph is the disjoint union of || copies of the corona  1 ∘ .Therefore, ( ∘ , ) = (( 1 ∘ , ))  .
As a consequence of the above theorem, we have the following corollary.
It is interesting that for the classification of graphs with exactly two, three, and four domination roots, we must consider some kinds of corona of two graphs.For more information, see [1].
To study more we need the following theorem.
Despite the above property, it is difficult to determine the domination polynomial of this product, even in such simple cases as the grid graphs   ◻  .Now we consider another operation of two graphs.Let  and  be graphs, with () = {V 1 , . . ., V  }.The graph  ⬦  formed by substituting a copy of  for every vertex of  is formally defined by taking a disjoint copy of ,  V , for every vertex V of  and joining every vertex in   to every vertex in  V if and only if  is adjacent to V in .
The following result is also proven in [6, Lemma 3].We would like to obtain some corollaries.We recall the following theorems.
Here we consider the graphs obtained by selecting one vertex in each of  triangles and identifying them.Some call them Dutch Windmill graphs [8,9] Theorem 8 can be used to generalize recurrence relations for the domination polynomial of some families of graphs.For example, we state and prove the following theorem.(iii) From Theorem 8, we have (  3 ⬦  , ) = (  3 , (1+ )  − 1).Now by Theorem 10, we have the result.

Conclusion
In this paper, we studied the domination polynomials of some graph operations.There are some open problems which are interesting to consider.
(i) What is the basic formula for the domination polynomial of the Cartesian product of two graphs?
For two graphs  and , let [] be the graph with vertex set () × () and such that vertex (, ) is adjacent to vertex (, ) if and only if  is adjacent to  (in ) or  =  and  is adjacent to  (in ).The graph [] is the lexicographic product (or composition) of  and  and can be thought of as the graph arising from  and  by substituting a copy of  for every vertex of .There is a main problem.
(ii) How can compute the domination polynomial of Lexicographic product of the two graphs?
8. For any graph , (⬦  , ) = (, (1+)  −1).Note that the closed neighborhood of the vertex (, V) of graph  ⬦   is (  [],   ).To make a dominating set of  ⬦   , suppose that  ⊂ () is a dominating set for .It is easy to see that ⋃ ∈ {(, V) | V ∈   } is a dominating set of  ⬦   , where {  |  ∈ } is a family of arbitrary nonempty subsets of (  ).Therefore, every V ∈  corresponds to all nonempty subsets of  ⬦   which have the generating function (1 + )  − 1.So we have the result.
and friendship graphs.Proof.It is easy to see that   3 is join of  1 and  2 .Now by Theorem 1, we have + (1 + ) 2 .