Spectral expansion properties of pseudorandom bipartite graphs

Abstract: An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have nice pseudorandom and expansion properties when the second largest adjacency eigenvalue is not large. In this paper, we prove several explicit properties of biregular bipartite graphs from spectral perspectives. In particular, we show that for any $(a,b)$-biregular bipartite graph $G$, if the spectral gap is greater than $\frac{2(k-1)}{\sqrt{(a+1)(b+1)}}$, then $G$ is $k$-edge-connected; and if the spectral gap is at least $\frac{2k}{\sqrt{(a+1)(b+1)}}$, then $G$ has at least $k$ edge-disjoint spanning trees. We also prove that if the spectral gap is at least $\frac{(k-1)\max{a,b}}{2\sqrt{ab - (k-1)\max{a,b}}}$, then $G$ is $k$-connected for $k\ge 2$; and if the spectral gap is at least $\frac{6k+2\max{a,b}}{\sqrt{(a-1)(b-1)}}$, then $G$ has at least $k$ edge-disjoint spanning 2-connected subgraphs. We have stronger results in the paper.