Growing Well-connected Graphs

• Published in: Proceedings of the 45th IEEE Conference on Decision and Control pp. 6605 - 6611
• Main Authors: Ghosh, A., Boyd, S.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.12.2006
• Subjects: Control systems; Eigenvalues and eigenfunctions; Information systems; Joining processes; Laboratories; Laplace equations; Robust stability; Upper bound; USA Councils; Vectors
• ISBN: 9781424401710; 1424401712
• ISSN: 0191-2216
• DOI: 10.1109/CDC.2006.377282

The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how well-connected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult combinatorial optimization, so we seek a heuristic for approximately solving the problem. The standard convex relaxation of the problem can be expressed as a semidefinite program (SDP); for modest sized problems, this yields a cheaply computable upper bound on the optimal value, as well as a heuristic for choosing the edges to be added. We describe a new greedy heuristic for the problem. The heuristic is based on the Fiedler vector, and therefore can be applied to very large graphs