Graph Operations and Neighbor Rupture Degree

In a communication network, the vulnerability parameters measure the resistance of the network to disruption of operation after the failure of certain stations or communication links. A vertex subversion strategy of a graphG, say S, is a set of vertices inGwhose closed neighborhood is removed from G. The survival subgraph is denoted by G/S. The neighbor rupture degree of G, Nr(G), is defined to be Nr(G) = max{w(G/S)−|S|−c(G/S) : S ⊂ V(G), w(G/S) ≥ 1}, where S is any vertex subversion strategy ofG,w(G/S) is the number of connected components in G/S and c(G/S) is the maximum order of the components of G/S (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.


Introduction
A network can be broke down completely or partially with unexpected reasons.If the data is not transmitted to the desired location that means there is a problem on the system.This problem can block a treaty of billions of euros or make a big problem for human's life.In these days the reliability and the vulnerability of networks are so important.For that reason graphs are taken as a model in the research area of reliability and vulnerability of the networks.Each network center is taken as a vertex and the connections of these vertices are edges of a graph.
A few questions can be asked at this point How can the reliability and the vulnerability of a network be determined?What are the factors of the reliability and the vulnerability?For example, what can be done if there is a problem on the way you are using every day to work?We have two choices; we may give up going to work although we have the risk of dismissal or we can look for another way to work.The question "if there is another way to reach work" may come to our minds.In other words "Has the link connection between home and work completely broken down?".To answer this question, we must know the dimensions of the problem between home and work.The vulnerability of the graph which represents the way between home and work should be searched.In graph theory some vulnerability parameters are defined to measure the vulnerability value of graphs such as connectivity [1], integrity [2], neighbor integrity [3], rupture degree [4], and neighbor rupture degree [5].
Terminology and notation not defined in this paper can be found in [5].Let  be a simple graph and let  be any vertex of .The set () = {V ∈ () | V ̸ = ; V and  are adjacent} is the open neighborhood of , and [] = {} ∪ () is the closed neighborhood of .A vertex  in  is said to be subverted if the closed neighborhood of  is removed from .A set of vertices  = { 1 ,  2 , . . .,   } is called a vertex subversion strategy of  if each of the vertices in  has been subverted from .If  has been subverted from the graph , then the remaining graph is called survival graph, denoted by /.

Basic Results
In this paper the new vulnerability parameter neighbor rupture degree was studied.The concept of neighbor rupture degree was introduced by Bacak-Turan and Kırlangıc in 2011 [5].The definition of neighbor rupture degree and some results are given below.
Definition 1 (see [6]).The neighbor rupture degree of a noncomplete connected graph  is defined to be where  is any vertex subversion strategy of , (/) is the number of connected components in /, and (/) is the maximum order of the components of /.
In particular, the neighbor rupture degree of a complete graph   is defined to be Nr Some known results are listed below.
Theorem 2 (see [6]).(a) Let   be a path graph with n vertices and  ≥ 2, (b) Let   be a cycle graph with n vertices and  ≥ 3,

Graph Operations and Neighbor Rupture Degree
In this section some graph operations are operated on graphs and their neighbor rupture degrees are evaluated.
Definition 3 (see [7]).The union graph  = Proof.Let  be a subversion strategy of Since these are complete graphs, it is obvious that  contains at most one vertex from each From ( 10) and ( 11) we obtain Nr( The following theorem's proof is very similar to that of Theorem 5. Theorem 6.Let   1 ,   2 ,   3 , . . .,    be complete graphs with  1 ≤  2 ≤  3 ⋅ ⋅ ⋅ ≤   where  +1 −   ≤ 2; for all  ∈  + .Then Corollary 7. Let Definition 8 (see [7]).The join graph In this part, neighbor rupture degree of join of some graphs is given.Theorem 9. Let  1 and  2 be two connected graphs.Then Proof.Let  be a subversion strategy of  1 +  2 .There are three cases according to the elements of .
For three or more disjoint graphs  1 ,  2 ,  3 , . . .,   sequential join The following theorem's proof is very similar to that of Theorem 9.
Theorem 10.Let  1 ,  2 , and  3 be connected graphs.Then the neighbor rupture degree of sequential join of  1 ,  2 , and  3 is Definition 13 (see [9]).The complement of a simple graph  is obtained by taking the vertices of  and joining two of them whenever they are not joined in  and denoted by   .
Theorem 14.Let   be a path graph of order .Then Proof.Let  be a subversion strategy of    and let  = {} where  ∈ (  ).
On the other hand, if we assume  is a subversion strategy with || ≥ 2, then the remaining graph is empty.Therefore it contradicts to the definition of neighbor rupture degree.From ( 21) and ( 22) we have Nr(   ) = −1.The following theorem's proof is very similar to that of Theorem 14.
Theorem 15.Let  1, be a wheel graph of order  + 1.Then Theorem 16.Let  , be a complete bipartite graph.Then Proof.It is obvious that   , =   ∪   .According to Corollary 7 we get the result.
There are two cases according to the number of elements in .Let () = 3 − 2 − 1.Since   < 0,  is a decreasing function, so it takes its maximum value at  = .
The following theorems' proofs are very similar to that of Theorem 19.
We have two cases according to the cardinality of .Definition 23 (see [9]).The tensor product  1 ⊗  2 of two simple graphs  1 and  2 is the graph with ( 1 ⊗  2 ) =  1 ×  2 and where in ( 1 ,  2 ) and Theorem 24.Let  3 ⊗   be a tensor product of  3 and   and  ≡ 0 (mod 4).Then Proof.Let  be a subversion strategy of  3 ⊗   and || =  be the number of removing vertices from  3 ⊗   .There are two cases according to the number of elements in .
The following theorem's proof is very similar to that of Theorem 24.
The vertices (,   ) are not adjacent to each other, neither do the vertices (  , ).But these are adjacent to each other, so Definition 27 (see [10]).
Let  be a subversion strategy of   [].We have two cases according to the elements of .(64) Definition 30 (see [11]).An th power of a graph  is formed by adding an edge between all pairs of vertices of  with distance at most .If  = 2 then it is called a second power of a graph also called a square.(72)

Conclusion
In this study, we investigate the neighbor rupture degree of graphs obtained by graph operations.The graph operations are used to obtain new graphs.Union, join, complement, composition, power, cartesian product, and tensor product are taken into consideration in this work.These operations are performed to various graphs and their neighbor rupture degrees were determined.

Case 1 .Case 2 .
If deg() = 1 in   , then  is adjacent to all vertices in    except its neighbor in   .It means |[]| =  − 1 in    , If deg() = 2 in   , then  is adjacent to all vertices in    except its neighbors in   .It means |[]| =  − 2 in    where the remaining two vertices are adjacent.Therefore,