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?

$\begingroup$
$\endgroup$

1
my(well-intentioned) 'fault' that this duplication happened. $\endgroup$