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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 783610, 5 pages
http://dx.doi.org/10.1155/2013/783610
Research Article

Edge-Neighbor-Rupture Degree of Graphs

Turgutlu Vocational Training School, Celal Bayar University, Turgutlu, 45400 Manisa, Turkey

Received 22 April 2013; Accepted 3 August 2013

Academic Editor: Frank Werner

Copyright © 2013 Ersin Aslan. 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

The edge-neighbor-rupture degree of a connected graph is defined to be , where is any edge-cut-strategy of , is the number of the components of , and is the maximum order of the components of . In this paper, the edge-neighbor-rupture degree of some graphs is obtained and the relations between edge-neighbor-rupture degree and other parameters are determined.

1. Introduction

In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. To measure the vulnerability we have some parameters which are connectivity [1], integrity [2], scattering number [3], and rupture degree [4].

A spy network can be modeled by a graph whose vertices represent the stations and whose edges represent the lines of communication. If a station is destroyed, the adjacent stations will be betrayed so that the betrayed stations become useless to network as a whole [5]. Therefore, instead of considering the stability of a communication network in standard sense, some new graph parameters such as vertex-neighbor-connectivity [6] and edge-neighbor-connectivity [7], vertex-neighbor-integrity [8] and edge-neighbor-integrity [9], vertex-neighbor-scattering number [10] and edge-neighbor-scattering number [11], and vertex-neighbor-rupture degree [12] were introduced to measure the stability of communication networks in “neighbor” sense.

We use Bondy and Murty [1] for terminology and notation not defined here and consider only finite simple connected graphs. Let be a graph and any edge in . The diameter of  , denoted by , is the maximum distance over all pairs of vertices in .

and are adjacent} is the open-edge-neighborhood of , and is the closed-edge-neighborhood of . An edge in is said to be subverted when is deleted from . In other words, if , . A set of edges is called an edge subversion strategy of if each of the edges in has been subverted from . The survival subgraph is denoted by . An edge subversion strategy is called an edge-cut-strategy of if the survival subgraph is disconnected or is a single vertex or the empty graph [13].

The edge-neighbor-connectivity of , , is the minimum size of all edge-cut-strategies of . A graph is m-edge-neighbor-connected if [7].

The edge-neighbor-integrity of a graph , , is defined to be where is any edge subversion strategy of and is maximum order of the components of [9].

The edge-neighbor-scattering number of , , is defined as where is any edge-cut-strategy of and is the number of the components of [11].

The known parameters concerning the neighborhoods do not deal with the number of the removing edges, the number of the components, and the number of the vertices in the largest component of the remaining graph in a disrupted network simultaneously. In order to fill this void in the literature, the current study proposes a definition of edge-neighbor-rupture degree which is a new parameter concerning these three values. Additionally, this study also analyzes the relations between edge-neighbor-rupture degree and some other parameters and obtains edge-neighbor-rupture degree of some graphs.

The edge-neighbor-rupture degree of a connected graph is defined to be where is any edge-cut-strategy of , is the number of the components of , and is the maximum order of the components of . A set is said to be the -set of if .

The edge-neighbor-rupture degree differs from edge- neighbor-connectivity, edge-neighbor-integrity, and edge-neighbor-scattering number in showing the vulnerability of networks. For example, consider the graphs and in Figure 1.

fig1
Figure 1: The graphs and .

It can be easily seen that the edge-neighbor-connectivity, edge-neighbor-integrity, and edge-neighbor-scattering number of these graphs are equal:

On the other hand, the edge-neighbor-rupture degrees of and are different:

Hence, the edge-neighbor-rupture degree is a better parameter for distinguishing vulnerability of graphs and .

2. Bounds for Edge-Neighbor-Rupture Degree

In this section some lower and upper bounds are given for the edge-neighbor-rupture degree of a graph using different graph parameters.

Theorem 1. Let be a connected graph of order . Then,

Proof. Let be an edge-cut-strategy of and . If , then and . Therefore,
Hence we have
The proof is completed.

Theorem 2. Let be a connected graph of order , and let , be the independent number and edge-neighbor-connectivity of , respectively. Then,

Proof. Let be an edge-cut-strategy of . For any of , , , and . Hence we get
The proof is completed.

Theorem 3. Let be a connected graph of order . If ; then .

Proof. Assume that ; then contains a path . Thus for any edge in , , , and for any two edges and in , , . Therefore a contradiction. Hence .
The proof is completed.

Theorem 4. Let be a connected graph of order and edge independence number of . Then,

Proof. Let be an edge-cut-strategy of . If , then contains isolated vertices and . From the definition of edge neighbor rupture degree we have
The proof is completed.

3. Edge-Neighbor-Rupture Degree of Some Graphs

In this section, we consider the edge-neighbor-rupture degree of some graphs.

Theorem 5. Let be a path with order . Then

Proof. Let be an edge-cut-strategy of and . If , then and . Thus the function takes its maximum value at , and we get where and where . So, we have
On the other hand, if , then we have and . Hence
It can be easily seen that there is an edge-cut-strategy of such that , +1, and where and where . Therefore,
The proof is completed by (16), (17), and (18).

Theorem 6. Let be a cycle with order . Then

Proof. Let be an edge-cut-strategy of and . If , then and . Thus the function takes its maximum value at , and we get where and where . So, we have
On the other hand, if , then we have and . Hence
It can be easily seen that there is an edge-cut-strategy of such that , and where and where . Therefore
The proof is completed by (21), (22), and (23).

Lemma 7 (see [7]). For any graph with order , .

Theorem 8. Let be a complete graph with order . Then

Proof. Let be an edge-cut-strategy of and . By Lemma 7 we know . If , then and . Hence,
It can be easily seen that there is an edge set of such that , then we have and . From the definition of edge-neighbor-rupture degree we have
The proof is completed.

Definition 9. We also call a star with vertices. Let be a double star with end-vertices, where and , and a common edge , as shown in Figure 2. Note that if either or is 0, then the double star is a star.

783610.fig.002
Figure 2: .

Theorem 10. Let be a tree of order . Then if and only if is either a star or a double star , where , , and .

Proof. If is a tree of order and , then by Theorem 3 we have either or . If , then   is a star . If , then is a double star , where , , and .
Conversely, let be either a star with the order or a double star , where , , and .
If is an edge-cut-strategy of and , then we have isolated vertices. Therefore we have and . So
It can be easily seen that there is an edge-cut-strategy of such that ; then we have and . So
If is an edge set of and , then we have and .
The proof is completed.

Theorem 11. Let be a complete bipartite graph with . Then

Proof. Assume . Let be an edge-cut-strategy of and . If , then and . Hence, the function is a decreasing function and takes its maximum value at , and we get
It can be easily seen that there is an edge set of such that ; then we have and . From the definition of edge neighbor rupture degree we have
Similarly, we obtain when .
Finally, we have
The proof is completed.

Definition 12. The wheel graph with spokes, , is the graph that consists of an n-cycle and one additional vertex, say , that is adjacent to all the vertices of the cycle. In Figure 3, we display .

783610.fig.003
Figure 3: The wheel graph .

Theorem 13. Let be a wheel graph with order . Then

Proof. The graph has subgraphs and . Let be any one edge of .
If and , then we get . So,
If and , then and . Hence,
If and , then and . Thus,
The proof is completed by (35), (36), and (37).

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