561 research outputs found
On the second eigenvalue of random bipartite biregular graphs
We consider the spectral gap of a uniformly chosen random
-biregular bipartite graph with , where
could possibly grow with and . Let be the adjacency matrix
of . Under the assumption that and we show
that with high probability. As a corollary,
combining the results from Tikhomirov and Youssef (2019), we confirm a
conjecture in Cook (2017) that the second singular value of a uniform random
-regular digraph is for with high
probability. This also implies that the second eigenvalue of a uniform random
-regular digraph is for with high
probability. Assuming and , we further prove that for a
random -biregular bipartite graph,
for all with
high probability. The proofs of the two results are based on the size biased
coupling method introduced in Cook, Goldstein, and Johnson (2018) for random
-regular graphs and several new switching operations we defined for random
bipartite biregular graphs.Comment: 37 pages, 3 figures. Corollary 1.4 added, a few typo fixe
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