On middle cube graphs

TL;DR

This paper investigates the properties of middle cube graphs, a family related to hypercubes, focusing on their spectral characteristics and their relation to distance-regular graphs.

Contribution

It provides a complete spectral characterization of middle cube graphs and explores their connection to odd graphs within the framework of distance-regular graphs.

Findings

Eigenvalues and multiplicities are explicitly determined.

Middle cube graphs are shown to be related to odd graphs by doubling vertices.

The spectral properties reveal insights into their structure and regularity.

Abstract

We study a family of graphs related to the $n$-cube. The middle cube graph of parameter $k$ is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).