Abstract
For a vertex of a graph , the lower connectivity, denoted by , is the smallest number of vertices that contains and those vertices whose deletion from produces a disconnected or a trivial graph. The average lower connectivity denoted by is the value . It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.
1. Introduction
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. The best known and most useful measures of how well a graph is connected is the connectivity, defined to be the minimum number of vertices in a set whose deletion results in a disconnected or trivial graph. As the connectivity is the worst-case measure, it does not always reflect what happens throughout the graph. Recent interest in the vulnerability and reliability of networks (communication, computer, and transportation) has given rise to a host of other measures, some of which are more global in nature; see, for example, [1, 2].
Let be a finite simple graph with vertex set and edge set . In the graph , denotes the number of vertices. The minimum degree of a graph is denoted by . A subset of vertices is a dominating set if every vertex in is adjacent to at least one vertex of . The domination number is the minimum cardinality of a dominating set. A subset of is called an independent set of if no two vertices of are adjacent to . An independent set is maximum if has no independent set with . The independence number of , , is the number of vertices in a maximum independent set of .
Henning [3] introduced the concept of average independence and average domination. For a vertex of a graph , the lower independence number, denoted by , is the minimum cardinality of a maximal independent set of that contains , and the lower domination number, denoted by , is the minimum cardinality of a dominating set of that contains . The average lower independence number of , denoted by , is the value and the average lower domination number of , denoted by , is the value . Since holds for every vertex , we have for any graph . Also, it is clear that and so and .
The ()-connectivity of , denoted by , is defined to be the maximum value of for which and are -connected. It is a well-known fact that the connectivity equals .
In 2002, Beineke et al. [4] introduced a parameter to give a more refined measure of the global “amount” of connectivity. If the order of is , then the average connectivity of , denoted by , is defined to be . The expression is sometimes referred to as the total connectivity of . Clearly, for any graph , .
There are a lot of researches on the connectivity of a graph [5]. Many works provide sufficient conditions for a graph to be maximally connected [6–8]. The average connectivity has been extensively studied [4, 9].
2. The Average Lower Connectivity of a Graph
We introduce a new vulnerability parameter, the average lower connectivity. For a vertex of a graph , the lower connectivity, denoted by , is the smallest number of vertices that contains and those vertices whose deletion from produces a disconnected or a trivial graph. We observe that(i);(ii) if and only if is a cut vertex;(iii) if and only if is complete.The average lower connectivity denoted by is the value , where will denote the number of vertices in graph and will denote the sum over all vertices of . For any graph , so . We also observe that(i);(ii) if and only if is complete.
Example 1. Let the graph be 3-cycle with one additional vertex and edge, as shown in Figure 1. It is easy to see that , , , and and we have .
Let and be graphs. Now one can ask the following question: is the average lower connectivity a suitable parameter? In other words, does the average lower connectivity distinguish between and ?
For example, consider the graphs in Figure 2.
(a)
(b)
It can be easily seen that the connectivity and average connectivity of these graphs are equal: On the other hand, the average lower connectivity of and is different: Thus, the average lower connectivity is a better parameter than the connectivity and average connectivity to distinguish these two graphs. The average parameters have been found to be more useful in some circumstances than the corresponding measures based on worst-case situations.
Theorem 2. Let be a connected graph. Then,
Proof. For every vertex of , . For at least one vertex , . Hence, the inequality is strict. Then, The proof is completed.
Theorem 3. Let be a connected graph. Then,
Proof. It is easy to see that . Therefore, So, we have The proof is completed.
Theorem 4. Let be a -connected and -regular graph. Then,
Proof. The cardinality of -sets is always the same for every vertex of any graph and equals . Then, we have This means that the proof is completed.
It is obvious that we can give the following equality for the average lower connectivity of the cycle .(i)The average lower connectivity of the cycle is 2.
Theorem 5. Let be a connected graph. Then,
Proof. For every vertex of , . Thus, The proof is completed.
3. Average Lower Connectivity of Several Specific Graphs
In this section, we determine the average lower connectivity of several special graphs.
Theorem 6. Let be a tree with order . If has vertices with degree 1, then
Proof. Assume that has vertices with degree 1 and vertices with degree at least 2. Let vertices set of be where in the set contains vertices with degree 1; in the set contains vertices with degree at least 2. If , then . We have to repeat this process for vertices with degree 1. If , then . We have to repeat this process for vertices with degree at least 2. Thus, we have The proof is completed.
Corollary 7. The average lower connectivity of(a)the path is ;(b)the star is ;(c)the comet is .
Theorem 8. Let be a complete bipartite graph. Then
Proof. Let the partite sets of be and with and . We distinguish two cases.
Case 1. If , then by Theorem 4 we have .
Case 2 . For , a minimum disconnecting set of that contains must be , so . On the other hand, for , a minimum disconnecting set of that contains must be , so . Elementary computation yields the result.
Definition 9. The wheel graph with spokes, , is the graph that consists of an ()-cycle and one additional vertex, say , that is adjacent to all the vertices of the cycle. In Figure 3, we display .
Theorem 10. Let be a wheel graph. Then,
Proof. The wheel graph has vertices. The cardinality of -sets is always the same for every vertex of any and equals 3. Then, we have This means that the proof is completed.
Definition 11. The gear graph is a wheel graph with a vertex added between each pair adjacent to graph vertices of the outer cycle. The gear graph has vertices and edges. In Figure 4 we display .
Theorem 12. Let be a gear graph. Then,
Proof. Let vertices set of be where in the set contains 1 vertex with degree , the set contains vertices with degree 2, and the set contains vertices with degree 3.If , then . If , then . Therefore, The proof is completed.
Now we give the definition of Cartesian product.
Definition 13. The Cartesian product of graphs and has as its vertex set and () is adjacent to () if either and is adjacent to or and is adjacent to .
Connectivity of graph products has already been studied by different authors. In [10] it is proved that .
Theorem 14. Let and be two connected graphs; then
Proof. We know . Therefore, we get The proof is completed.
Proposition 15. For positive integer ,(i); (ii).
Proposition 16. Let and be positive integers. Then
Conflict of Interests
The author declares that there is noconflict of interests regarding the publication of this paper.
Acknowledgment
The author wishes to thank Bo-Qing Dong for his helpful suggestions and corrections.