Bisection algorithm of increasing algebraic connectivity by adding an edge

For a given graph (or network) G, consider another graph G´ by adding an edge e to G. We propose a computationally efficient algorithm of finding e such that the second smallest eigenvalue (algebraic connectivity, lambda₂(G´)) of G´ is maximized. Theoretically, the proposed algorithm runs in O(4 mnlog(d/isin)), where n is the number of nodes in G, m is the number of disconnected edges in G, d is the difference between lambda₃(G) and lambda₂(G), and isin > 0 is a sufficiently small constant. However, extensive simulations show that the practical computational complexity of the proposed algorithm, O(5.7 mn), is nearly comparable to that of a simple greedy-type heuristic, O(2 mn). This algorithm can also be easily modified for finding e which affects lambda₂(G) the least.