Ela on the Spectra of Johnson Graphs

The spectrum of a Johnson graph is known to be given by the Eberlein polynomial. In this paper, a straightforward representation-theoretic derivation of this fact is presented. Also discussed are some consequences of this formula, such as the fact that infinitely many of them are Ramanujan. 1. Introduction. Johnson graphs are heavily studied objects. The spectrum of any graph—that is, the multiset of eigenvalues of its adjacency matrix—is an important invariant from which much information about the graph can be ascertained. It is known that the spectrum of a Johnson graph is given by the Eberlein polynomial. While some of the above papers prove this result in a representation theoretic way, in this paper, we present an alternate proof, one which combines combinatorial and representation-theoretic techniques.