# spectrum of orthogonality graphs

The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard inner product space $\mathbb{Q}^n$). Since two such vectors are orthogonal if and only if the number of components of like sign is exactly equal to $n/2$, this graph is isomorphic to the graph with vertex set $\lbrace0,1\rbrace^n$ and two vertices adjacent if and only if their Hamming distance is equal to $n/2$. I am seeking the spectrum of $\Omega(n)$ - not just the eigenvalues but their multiplicities. In particular I am looking for the inertia of these graphs - that is the numbers of positive, zero and negative eigenvalues. Does anyone know where I should look?

• For completeness: an equivalent question was asked at mathoverflow.net/questions/294675/…, and in a sense it was my (well-intentioned) 'fault' that this duplication happened. Mar 9, 2018 at 14:59