The Domination Entropy of The Graphs

Graph entropies have been used to interpret the complexity of the networks. There are various graph entropies in the literature. In this note a new graph entropy is defined which is based on the dominating sets of the graphs. Moreover the entropy of some classes of the graphs is obtained


Introduction
There are many studies to determine the complexity of the networks in the last years. Computation of graph entropy measures is used in interdisciplinary researchs for example chemistry, information science and biology [1 − 6]. In order to determine the complexity of the graphs, most of the graph entropies are based on entropy of Shannon [7].
In the literature there are several graph entropy measures by using the order of the graphs, degree sequence of the graphs, distance of the graphs, characteristic polynomials and other graph polynomials of the graphs [7 − 14]. Graph entropies which are related to molecular descriptors are introduced in the last years. Calculation of graph entropy measures which are based on matchings and independent sets are defined [15] and furter studied [16]. Moreover some relations between comlexity of the graphs and Hosoya entropy are investigated Mowshowitz and Dehmer [17]. More details about the graph entropies can be found in the book [18].
As noted in [15] it could be useful to define new graph entropies based on different graph invariants. In this not a new graph entropy is introduced which is based on the dominating sets of the graphs and the entropy of some special graphs is calculated. In order to define this entropy, Dehmer's information functional approach [8] is used.

Preliminaries
Let be a simple graph with the vertex set ( ) and the edge set ( ). For a vertex ∈ ( ) the notation ( ) = { | ∈ ( )} denotes the vertices which are adjacent to and [ ] = { } ∪ ( ). The degree of a vertex is cardinality of ( ) and it is denoted by ( ).
A subset ⊆ ( ) is a dominating set, if every vertex of ∖ is adjacent to at least one vertex in . The notation ( ) is used to show the domination number of a graph and it is cardinality of a dominating set with minimum order [19]. Domination polynomials are introduced by Alikhani and Peng [20]. Moreover domination polynomials of paths [21], cycles [22] and caterpillar graphs [23] are studied. In order to characterize the graphs, using of domination polynomials is an useful way. In this paper we also use domination polynomials for determining the domination entropy of the graphs. Definition 1. The notation ( , ) is used to denote the family of dominating sets of with cardinality and the notation is used to denote the cardinality of ( , ) suc that ( ) = | ( , )|. Therefore the domination polynomial ( , ) of is introduced by the following equation [20] ( , ) = ∑ ( ) The entropy which are based on matchings and independent sets are computed by Hosoya and Merrifield-Simmons indices, respectively. Because total number of matchings is equal to Hosoya index and total number of independent sets is equal to Merrifield-Simmons index [15].
In here to denote the total number of dominating sets, we use instead of . is used to denote total domination number of the graphs [19] which is a domination parameter different from the total number of the dominating sets of the graphs.

Definition 2.
Let (G) be the total number of dominating sets of a graph . It is clear that is equal to the sum of the coefficients of the domination polynomials of the graph such that Definition 3. The entropy of a graph is defined by Dehmer's information functional approach [8] where an arbitrary information functional denoted by as follows Now we can give the definiton of the domination entropy by using a new information funtional. Then by using Definition 3 we obtain the domination entropy

Domination Entropy of Some Graphs
We first give an essential theorem about the entropy of a graph which is the union of two connected graphs.
Theorem 1. Assume that 1 and 2 are two connected graphs and = 1 ∪ 2 is the disjoint union of 1 and 2 . We obtain that Proof. Let be a complete graph . We know that domination polynomial of the is [20] ( , ) = −1 + (1 + ) −1 .
Now we can determine the number of the dominating sets with cardinality . The star graph have a vertex which has degree ( − 1) and the other ( − 1) vertices have degree one. In order to dominate the graph , the vertex is contained by every dominating set except that a set. The only dominating set of the which has cardinality ( − 1) does not contain and it is consisted of the vertices with degree one.
Therefore the set with cardinality one is { } and 1 ( ) = 1.
We determine the number of the dominating sets with cardinality . The star graph , have two vertices and which have degree and , respectively. The remaining ( + − 2) vertices have degree one. In order to dominate the graph , , there are three cases. These cases are, the vertices , are contained by dominating sets, one of the and is contained in dominating sets and none of the and is contained in dominating sets [23]. For the last case the only dominating set of the , does not contain the vertices and and it is consisted of the vertices with degree one. Therefore the set with cardinality two is { , } and 2 ( ) = 1.
The sets with cardinality three are consisted of , and a vertex whose degree is one. Thus 3 ( ) = + − 2. By this way we obtain the dominating sets of the , such that We can generalize the cardinality of the dominating sets with cardinality ( ) for ( − 3) ≥ ≥ 2 such that 3 ≤ ≤ + − 2. We accept that ( ) = 0 for < .
Finally we obtain that Moreover the partite sets of , is consisted of two sets with vertices and vertices. Thus we use the domination number of , with ( , ) =2 with chosen vertices, a vertex fromvertices and a vertex from -vertices. By Definition 2 we obtain that Assume that ≤ . Now we determine the number of dominating sets of the , such that

Conclusion
In this paper we defined domination entropy by using information functionals with the dominating sets of the graphs. In future works we compare the domination entropy of the graphs with the entropies based on the mathchings and independent sets. Furthermore total number of matchings and total number of independent sets are well known topological indices in graph theory as Hosoya index and Merrifield-Simmons index. The total number of dominating sets is not studied as a topological index to our best knowledge. It is an interesting topic, investigation of the number of dominating sets of the graphs which can be called as 'Domination Index'.