How hard is it to marry at random? (On the approximation of the permanent)
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Combinatorica
Better expanders and superconcentrators
Journal of Algorithms
The token distribution problem
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Random Cayley Digraphs and the Discrete Logarithm
ANTS-V Proceedings of the 5th International Symposium on Algorithmic Number Theory
Eigenvalues and graph bisection: An average-case analysis
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Undirected connectivity in log-space
Journal of the ACM (JACM)
Classical and quantum tensor product expanders
Quantum Information & Computation
Jellyfish: networking data centers randomly
NSDI'12 Proceedings of the 9th USENIX conference on Networked Systems Design and Implementation
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Expanders have many applications in Computer Science. It is known that random d-regular graphs are very efficient expanders, almost surely. However, checking whether a particular graph is a good expander is co-NP-complete. We show that the second eigenvalue of d-regular graphs, λ2, is concentrated in an interval of width O(√d) around its mean, and that its mean is O(d3/4). The result holds under various models for random d-regular graphs. As a consequence a random d-regular graph on n vertices, is, with high probability a certifiable efficient expander for n sufficiently large. The bound on the width of the interval is derived from martingale theory and the bound on E(λ2) is obtained by exploring the properties of random walks in random graphs.