Related the Connectivity of a Graph with its Complement

There are a lot of papers discussing the connectivity of a graph and its complement graph. For example, it has been characterized that a graph G and its complement G(bar) are both have connectivity one or both are 2-connected. In this paper, we establish a simple formula to compute the connectivity of a graph that comes from the join of graphs. Based on this formula, we derive an upper bound and a lower bound to the connectivity of the complement of a graph. We also show that if a k-connected graph G fulfills a certain condition related to its complement, then G is k-geodetically connected.