The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the adjacency matrix of the graph but with $\pm 1$ entries denoting which edges got lifted by the identity("+") permutation on two elements and which ones got the flip("-")

Now one knows that for such a $2-$lift the adjacency eigenvalues of the new lifted graph is the union of the old eigenvalues and the eigenvalues of the signing matrix.

- I want to know as to how/if a positive resolution to the Bilu-Linial conjecture necessarily will imply that this special $2-$lift can be used to construct Ramanujan graphs.

Even if this special signing claimed by Bilu-Linial were to actually exist wouldn't it still be possible that may be the original graph had eigenvalues in the interval $(2\sqrt{d-1},d)$ and which would still persist and hence the lifted graph wouldn't become Ramanujan? Or am I missing something?

- Also what is the significance of the lower bound of $-2\sqrt{d-1}$ in the Bilu-Linial conjecture? (the definition of the Ramanujan graph doesn't seem to need that..)

By a "Ramanujan graph" I would mean a $d-$regular graph whose second largest eigenvalue is $\leq 2\sqrt{d-1}$.

I am aware of the proof of the positive resolution of this conjecture for the bipartite case by Nikhil Srivastava, Adam Marcus and Daniel Spielman. (..though their proof doesn't seem to use bipartition anywhere - its just that they can control for only the upper bound $2\sqrt{d-1}$ and the lower bound gets implied by the bipartition structure and hence the theorem holds..)