Ramanujan Coverings of Graphs
Chris Hall, Doron Puder, and William F. Sawin

TL;DR
This paper proves the existence of Ramanujan r-coverings for graphs, generalizing previous results, and introduces new polynomial tools and group labelings to construct richer families of Ramanujan graphs.
Contribution
It establishes the existence of Ramanujan r-coverings for graphs using interlacing families and extends the concept to group labelings, broadening the class of Ramanujan graphs.
Findings
Existence of r-coverings with bounded eigenvalues
Generalization of matching polynomial to r-coverings
Broader group labelings for Ramanujan coverings
Abstract
Let be a finite connected graph, and let be the spectral radius of its universal cover. For example, if is -regular then . We show that for every , there is an -covering (a.k.a. an -lift) of where all the new eigenvalues are bounded from above by . It follows that a bipartite Ramanujan graph has a Ramanujan -covering for every . This generalizes the case due to Marcus, Spielman and Srivastava (2013). Every -covering of corresponds to a labeling of the edges of by elements of the symmetric group . We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by…
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