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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 836395, 7 pages
http://dx.doi.org/10.1155/2013/836395
Research Article

Graph Operations and Neighbor Rupture Degree

1Department of Mathematics, Yasar University, 35100 Izmir, Turkey
2Department of Mathematics, Celal Bayar University, 45140 Manisa, Turkey

Received 25 March 2013; Revised 13 June 2013; Accepted 14 June 2013

Academic Editor: Frank Werner

Copyright © 2013 Saadet Kandİlcİ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 graph , say , is a set of vertices in whose closed neighborhood is removed from . The survival subgraph is denoted by . The neighbor rupture degree of , , 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 (G. Bacak Turan, 2010). In this paper we give some results for the neighbor rupture degree of the graphs obtained by some graph operations.

1. 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 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 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 .

2. 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 . A set is said to be of if Some known results are listed below.

Theorem 2 (see [6]). (a) Let be a path graph with n vertices and ,
(b) Let be a cycle graph with n vertices and ,
(c) Let be a -partite graph
(d) Let be a wheel graph with 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 has vertex set and edge set . If a graph consists of disjoint copies of a graph , then we write .

Theorem 4. Let be connected graphs. Then

Proof. Let be the union of . Let be -sets of respectively, and let be a subversion strategy of . Then we obtain Thus we have .

Theorem 5. Let be complete graphs with where . Then

Proof. Let be a subversion strategy of . Since these are complete graphs, it is obvious that contains at most one vertex from each .
If , then and . Thus we have since There exist such that , and . Then we have From (10) and (11) we obtain .

The following theorem’s proof is very similar to that of Theorem 5.

Theorem 6. Let be complete graphs with where ;  for all . Then

Corollary 7. Let

Definition 8 (see [7]). The join graph has vertex set and edge set .

In this part, neighbor rupture degree of join of some graphs is given.

Theorem 9. Let and be two connected graphs. Then

Proof. Let be a subversion strategy of . There are three cases according to the elements of .
Case 1. Let be the -set of such that . Since any elements from are adjacent to every element of in , we have
Case 2. Let be the -set of such that . Since any elements from are adjacent to every element of in , we have
Case 3. Let . Since contains at least one vertex of which is adjacent to all the vertices of and contains at least one vertex of which is adjacent to all the vertices of in , then is empty. It contradicts to the definition of neighbor rupture degree.
By (15) and (16) .

For three or more disjoint graphs sequential join is the graph [8].

The following theorem’s proof is very similar to that of Theorem 9.

Theorem 10. Let , , and be connected graphs. Then the neighbor rupture degree of sequential join of  , , and is

Corollary 11. If , then

Corollary 12. If , then

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 .
Case 1.   If in , then is adjacent to all vertices in except its neighbor in . It means in , then we have
Case 2.   If in , then is adjacent to all vertices in except its neighbors in . It means in where the remaining two vertices are adjacent. Therefore, On the other hand, if we assume is a subversion strategy with , then the remaining graph is empty. Therefore it contradicts to the definition of neighbor rupture degree.
From (21) and (22) we have .

The following theorem’s proof is very similar to that of Theorem 14.

Theorem 15. Let be a wheel graph of order . Then

Theorem 16. Let be a complete bipartite graph. Then

Proof. It is obvious that .
According to Corollary 7 we get the result.

Corollary 17. Let be a star graph of order . Then

Definition 18 (see [7]). The cartesian product has , and two vertices and of are adjacent if and only if either or

Theorem 19. Let be a cartesian product with . Then

Proof. Let be a subversion strategy of and . There are two cases according to the number of elements in .
Case 1. Let . Then and .
Since and Let . is an increasing function since . So it takes its maximum value at . Then . Hence,
Case 2. Let . Then and , thus we obtain .
Let . Since , is a decreasing function, so it takes its maximum value at . Then From (30) and (31) we have It is obvious that there exist such that , and so From (32) and (33) we have .

The following theorems’ proofs are very similar to that of Theorem 19.

Theorem 20. Let be a cartesian product with . Then

Theorem 21. Let be a cartesian product with . Then

Theorem 22. Let and be two complete graphs with (). Then

Proof. Let be a subversion strategy of and let . We have two cases according to the cardinality of .
Case 1. If , then and , so we have Let . Since is an increasing function in , it takes its maximum value at and . Thus we get
Case 2. If , then and . So we have .
Let . Since is a decreasing function, it takes its maximum value at and . Thus we get From (38) and (39) we have There exist such that , and , thus we have From (40) and (41) we get .

Definition 23 (see [9]). The tensor product of two simple graphs and is the graph with and where in and are adjacent in if, and only if, is adjacent to in and is adjacent to in .

Theorem 24. Let be a tensor product of and   and . Then

Proof. Let be a subversion strategy of and be the number of removing vertices from . There are two cases according to the number of elements in .
Case 1. If , then and . Thus we have
Case 2. If , then and . Thus we have Let since the function is a decreasing function so it takes its maximum value at From (43) and (45) we get There exist such that , and thus we have From (46) and (47) we get .

The following theorem’s proof is very similar to that of Theorem 24.

Theorem 25. Let be a tensor product of and and . Then

Theorem 26. Let the tensor product of and is . Then

Proof. Let be any vertex of such that and . The only vertices that are not adjacent to in are with and with , where and .
The vertices are not adjacent to each other, neither do the vertices . But these are adjacent to each other, so

Definition 27 (see [10]). The composition of simple graphs and is the simple graph with vertex set , in which is adjacent if and only if either or and .

Theorem 28. Let be the composition of and with . Then neighbor rupture degree of is

Proof. Let the vertex set of be labeled as , , and . xy(52)
It can be easily seen that is the sequential join of three disjoint path graphs, . Then, according to the Theorem 10 we get By Theorem 2, we have
To conclude the proof we need to find . Let be a subversion strategy of . Since has two identical disjoint path graphs, let denote the number of removing vertices of each and let .
Case 1. If and , then we get Let , since is an increasing function it takes its maximum value at . . Since neighbor rupture degree has to be an integer, we have and we get
Case 2. If , , and , thus we get From (56) and (57) we have There exist such that , and for , for .
Thus we have From (58) and (59) we get By (54) and (60) we conclude the proof.

Theorem 29. Neighbor rupture degree of composition of and any graph is

Proof. Let the vertex set of be labeled as, , xy(62) Let be a subversion strategy of . We have two cases according to the elements of .
Case 1. Let we choose one element from any vertex set ; if and , then it removes all of other vertex sets. So it depends on only which we choose one element. Then we have
Case 2. Let we choose two elements from any vertex set and with . Then is empty set. It contradicts to the definition of neighbor rupture degree. Thus we obtain

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 then it is called a second power of a graph also called a square.

Theorem 31. Neighbor rupture degree of is

Proof. Let be a subversion strategy of and let . There are two cases according to the number of elements of .
Case 1. If , then , ; then we get Let
Case 2. If , then ; then we get
Case 3. If , then , ; then we get According to (67), (68), and (69) we have There exist such that , , ; then By (70) and (71) we get the result,

4. 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.

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