On the average lower bondage number of a graph

The domination number is an important subject that it has become one of the most widely studied topics in graph theory, and also is the most often studied property of vulnerability of communication networks. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. Let G = ( V ( G ) ,E ( G )) be a simple graph. The bondage number b ( G ) of a nonempty graph G is the smallest number of edges whose removal from G result in a graph with domination number greater than that of G . If we think a graph as a modeling of network, the average lower bondage number of a graph is a new measure of the graph vulnerability and it is defined by b av (G) = 1/| E(G) | ∑ e ∈ E(G) b e (G) , where the lower bondage number, denoted by b e ( G ), of the graph G relative to e is the minimum cardinality of bondage set in G that contains the edge e . In this paper, the above mentioned new parameter has been defined and examined. Then upper bounds, lower bounds and exact formulas have been obtained for any graph G . Finally, the exact values have been determined for some well-known graph families.