Ramanujan Coverings of Graphs

Understand

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover.

• For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$.
• We show that for every $r$, there is an $r$-covering (a.k.a.
• an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$.