The Average Lower Connectivity of Graphs

For a vertex V of a graph G, the lower connectivity, denoted by sV(G), is the smallest number of vertices that contains V and those vertices whose deletion from G produces a disconnected or a trivial graph. The average lower connectivity denoted by κav(G) is the value (∑V∈V(G) sV(G))/|V(G)|. 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.


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 V of a graph , the lower independence number, denoted by  V (), is the minimum cardinality of a maximal independent set of  that contains V, and the lower domination number, denoted by  V (), is the minimum cardinality of a dominating set of  that contains V.The average lower independence number of , denoted by  V (), is the value (∑ V∈()  V ())/|()| and the average lower domination number of , denoted by  V (), is the value (∑ V∈()  V ())/|()|.Since  V () ≤  V () holds for every vertex V, we have  V () ≤  V () for any graph .Also, it is clear that () = min{ V () | V ∈ ()} and () = min{ V () | V ∈ ()} so () ≤  V () and () ≤  V ().
The (, V)-connectivity of , denoted by   (, V), is defined to be the maximum value of  for which  and V are -connected.It is a well-known fact that the connectivity () equals min{  (, V), V ∈ ()}.
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 () = (∑ ,V   (, V))/ (  2 ).The expression ∑ ,V   (, V) 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][7][8].The average connectivity has been extensively studied [4,9].

The Average Lower Connectivity of a Graph
We introduce a new vulnerability parameter, the average lower connectivity.For a vertex V of a graph , the lower connectivity, denoted by  V (), is the smallest number of vertices that contains V and those vertices whose deletion from  produces a disconnected or a trivial graph.We observe that The average lower connectivity denoted by  V () is the value (∑ V∈()  V ())/, where  will denote the number of vertices in graph  and ∑ V∈()  V () will denote the sum over all vertices of .For any graph , () = min{ V () | V ∈ ()} so  av () ≥ ().We also observe that Let the graph  be 3-cycle with one additional vertex and edge, as shown in Figure 1.It is easy to see that   () = 2,   () = 2,   () = 1, and   () = 3 and we have Let  1 and  2 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  1 and  2 ?
For example, consider the graphs in Figure 2. 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  1 and  2 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 ,  V () ≤ ()+2.For at least one vertex V,  V () = ().Hence, the inequality is strict.Then, The proof is completed.
The proof is completed.

Theorem 4.
Let  be a -connected and -regular graph.Then, Proof.The cardinality of  V ()-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.

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 () = ( 1 )∪( 2 ) where in ( 1 ) the set contains  vertices with degree 1; in ( 2 ) the set contains  −  vertices with degree at least 2. If V ∈ ( 1 ), then  V ( 1 ) = 2.We have to repeat this process for  vertices with degree 1.If V ∈ ( 2 ), then  V ( 2 ) = 1.We have to repeat this process for  −  vertices with degree at least 2. Thus, we have The proof is completed.Case 2 ( < ).For  ∈ , a minimum disconnecting set of  that contains  must be  ∪ {}, so   () =  + 1.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  − 1 spokes,   , is the graph that consists of an ( − 1)-cycle and one additional vertex, say , that is adjacent to all the vertices of the cycle.
In Figure 3, we display  7 .
Theorem 10.Let   be a wheel graph.Then,  Proof.The wheel graph   has  vertices.The cardinality of  V ()-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 2+1 vertices and 3 edges.
In Figure 4 we display  6 .