Large regular bipartite graphs with median eigenvalue 1

A recent result of one of the authors says that every connected subcubic bipartite graph that is not isomorphic to the Heawood graph has at least one, and in fact a positive proportion of its eigenvalues in the interval [−1,1][−1,1]. We construct an infinite family of connected cubic bipartite graphs which have no eigenvalues in the open interval (−1,1)(−1,1), thus showing that the interval [−1,1][−1,1] cannot be replaced by any smaller symmetric subinterval even when allowing any finite number of exceptions. Similar examples with vertices of larger degrees are considered and it is also shown that their eigenvalue distribution has somewhat unusual properties. By taking limits of these graphs, we obtain examples of infinite vertex-transitive r  -regular graphs for every r⩾3r⩾3, whose spectrum consists of points ±1 together with intervals [r−2,r][r−2,r] and [−r,−r+2][−r,−r+2]. These examples shed some light onto a question communicated by Daniel Lenz and Matthias Keller with motivation in relation to the Baum–Connes conjecture.