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  1. I need the bound for the spectrum gap of random semiregular ($\ell$, $r$)-bipartite graph. This paper (http://arxiv.org/abs/1212.5216) gives the bound for $\ell$-regular bipartite graphs (with $\ell = r$). But are there similar results for $\ell \leq r$?

  2. Assume each edge $(ij)$ in the graph is assigned with a random variable $w_{ij}$, with finite mean and finite variance. What is the spectrum gap of the weighted adjacency matrix $W = [w_{ij}]$? Again assuming the graph is a random semiregular ($\ell$, $r$)-bipartite graph again.

Thank you very much!

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