On the second eigenvalue of random bipartite biregular graphs

TL;DR

This paper investigates the spectral gap of random bipartite biregular graphs, establishing bounds on the second eigenvalue and singular values under various degree growth conditions, using size biased coupling and switching techniques.

Contribution

It provides new bounds on the second eigenvalue of random bipartite biregular graphs and extends spectral analysis techniques to these graphs with growing degrees.

Findings

Second eigenvalue is O(√d₁) when d₂=O(n^{2/3})

Second singular value of d-regular digraphs is O(√d) for 1≤d≤n/2

Eigenvalues are tightly concentrated around d₁ with deviations of O(√d₁)

Abstract

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 $d_1\geq d_2$ 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 showed that the second singular value of a uniform random $d$-regular digraph is $O(\sqrt{d})$ for $1\leq d\leq n/2$ with high probability. Assuming $d_2$ is fixed and $d_1=O(n^2)$, we further prove that for a random $(d_1,d_2)$-biregular bipartite graph, $|\lambda_i^2(A)-d_1|=O(\sqrt{d_1})$ for all $2\leq i\leq n+m-1$ 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.