On the Spectrum of Middle-Cubes

TL;DR

This paper characterizes the spectrum of middle-cubes, a class of graphs related to the Middle Levels conjecture, using two different mathematical approaches, one leveraging Johnson graphs and the other purely graph-theoretic.

Contribution

It provides a complete spectral characterization of middle-cubes with two distinct proofs, enriching understanding of their algebraic structure.

Findings

Spectral characterization of middle-cubes established.

Connection to Johnson graphs and association schemes demonstrated.

Alternative pure graph theory proof provided.

Abstract

A middle-cube is an induced subgraph consisting of nodes at the middle two layers of a hypercube. The middle-cubes are related to the well-known Revolving Door (Middle Levels) conjecture. We study the middle-cube graph by completely characterizing its spectrum. Specifically, we first present a simple proof of its spectrum utilizing the fact that the graph is related to Johnson graphs which are distance-regular graphs and whose eigenvalues can be computed using the association schemes. We then give a second proof from a pure graph theory point of view without using its distance regular property and the technique of association schemes.