TITLE: Spectrum of orthogonality graph (2)
QUESTION [9 upvotes]: The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent. 
I am only interested when $4|n$, since otherwise $\Omega(n)$ is empty or bipartite.  I am keen to know the spectrum of $\Omega(n)$ - the eigenvalues and their multiplicities. In particular I am seeking the inertia of $\Omega(n)$ - that is the numbers of positive, zero and negative eigenvalues. Many thanks Clive

REPLY [3 votes]: The graph $\Omega(n)$ is one of the relations of the Hamming association scheme $H(n,2)$, namely, the one corresponding to the Hamming distance $n/2$, see e.g. here. Its eigenvalues are given by the values of the Krawchuk's polynomials $K_k(n/2)$, as $0\leq k\leq n$, see details in my answer to your MO question 295493,
and multiplicities are just $\binom{n}{k}$, $0\leq k\leq n$. 
For this one can e.g. compute that if $n$ is divisible by 4 then the number of 0 eigenvalues of $\Omega(n)$ is $2^{n-1}$.