Ramanujan coverings of graphs

Abstract

Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2 √ 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 ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r = 2 case due to Marcus, Spielman and Srivastava [MSS15a].Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group S r . 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 [MSS15a], a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from [MSS15b].Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [−ρ, ρ].

On the Expansion of Group-Based Lifts

“…This connection tells us that the lift graph incurs the eigenvalues of the base graph, while its new eigenvalues are the union of eigenvalues of a collection of matrices arising from the irreducible representations of the group and the group elements assigned to the edges. This connection has been recently used in [HPS15] in the context of expansion of lifts, aiming to generalize the results in [MSS15]. In order to understand the expansion properties of the lifts, we focus on the new eigenvalues of the lifted graph.…”

Finite free convolutions of polynomials

“…The papers [15][16][17] cite theorems in the original arxiv version of this work [6], which are numbered differently from the ones in this paper. In particular, Theorems 1.7 and 1.8 of [6] correspond to Theorems 1.12 and 1.14 of this paper.…”

“…Proof Note that equality in (17) clearly implies equality in (16). Now, assume by way of contradiction that ( 16) is false, and let x = maxroot ( f ).…”

A New [Combinatorial] Proof of the Commutativity of Matching Polynomials for Cycles

“…However, recently, in [5], Marcus, Spielman, and Srivastava proved that every bipartite Ramanujan graph has a Ramanujan 2-covering graph, which showed that there were an infinite number of Ramanujan graphs of any degree of regularity. In [3], Hall, Puder, and Sawin generalized this result by proving that every bipartite graph without self-loops has a Ramanujan d-covering for every d. To do so, they introduced the d-matching polynomial. Definition 1.1.1.…”

“…, d}, and G λ is the d-cover corresponding (in a sense which will be made precise in Section 2.3) to a given labeling λ ∈ L d (G). Essentially, the d-matching polynomial is the average of all of the (classical) matching polynomials that come from the d-covers of a graph G. It is known due to Heilmann and Leib in [4] that the matching polynomial has only real roots when G has no self-loops, and Hall et al showed that the d-matching polynomial shares this important property [3]. Independently, Hall made the following conjecture:…”