Matrix analysis
Constructing Expander Graphs by 2-Lifts and Discrepancy vs. Spectral Gap
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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Various parameters have been discovered which give a measurement of the "randomness" of a graph. We consider two such parameters for directed graphs: the singular values of the (normalized) adjacency matrix and discrepancy (a measurement of how randomly edges have been placed). We will show that these two are equivalent by bounding one by the other so that if one is small then both are small. We will also give a related result for discrepancy of walks when the indegree and out-degree at each vertex is equal. Both of these results follow from a more general discrepancy property of nonnegative matrices which we will state and prove.