Using discrepancy to control singular values for nonnegative matrices

We will consider two parameters which can be associated with a nonnegative matrix: the second largest singular value of the “normalized” matrix, and the discrepancy of the entries (which is a measurement between the sum of the actual entries in blocks versus the expected sum). Our main result is to show that these are related in that discrepancy can be bounded by the second largest singular value and vice versa. These matrix results are then used to derive some (edge/alternating walks) discrepancy properties of edge-weighted directed graphs.