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Register a new userOn the second eigenvalue of random bipartite biregular graphs
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Yizhe Zhu
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We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption that $d_1geq d_2$ and $d_2=O(n^{2/3}),$ we show that $lambda_2(A)=O(sqrt{d_1})$ 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 $d$-regular digraph is $O(sqrt{d})$ for $1leq dleq n/2$ with high probability. This also implies that the second eigenvalue of a uniform random $d$-regular digraph is $O(sqrt{d})$ for $1leq dleq n/2$ with high probability. Assuming $d_2=O(1)$ and $d_1=O(n^2)$, we further prove that for a random $(d_1,d_2)$-biregular bipartite graph, $|lambda_i^2(A)-d_1|=O(sqrt{d_1(d_2-1)})$ for all $2leq ileq n+m-1$ 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 $d$-regular graphs and several new switching operations we defined for random bipartite biregular graphs.
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