Bisection Algorithm of Increasing Algebraic Connectivity by Adding an Edge

For a given graph (or network) G, consider another graph G´ by adding or deleting an edge e to or from G. We propose a computationally efficient algorithm of finding e such that the second smallest eigenvalue (algebraic connectivity, ??₂(G´)) of G´ is maximized or minimized. Theoretically, the proposed algorithm runs in O(4mnlog(d/??)), where n is the number of nodes in G, m is the number of disconnected edges in G, d is the difference between ??₃(G) and ??₂(G), and ?? > 0 is a sufficiently small constant. However, extensive simulations show that the practical computational complexity of the proposed algorithm, O(5.7mn), is nearly comparable to that of a simple greedy-type heuristic, O(2mn). This algorithm can also be easily modified for finding e which affects ??₂(G) the least.