On middle cube graphs - C. Dalfo; M. A. Fiol; M. Mitjana (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2015)
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).