  We prove an analogue of Alon's spectral gap conjecture for random bipartite,
biregular graphs. We use the Ihara-Bass formula to connect the non-backtracking
spectrum to that of the adjacency matrix, employing the moment method to show
there exists a spectral gap for the non-backtracking matrix. A byproduct of our
main theorem is that random rectangular zero-one matrices with fixed row and
column sums are full-rank with high probability. Finally, we illustrate
applications to community detection, coding theory, and deterministic matrix
completion.
