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, λ<inf>2</inf>(G′)) of G′ is maximized. 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 λ<inf>3</inf>(G) and λ<inf>2</inf>(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 λ<inf>2</inf>(G) the least.

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