Consider the graph whose vertices are labelled by the binary representation of the integers from $0$ to $2^{d}-1$ for some $d \in \mathbb{N}$. So its a graph with $2^d$ vertices. An edge exists between two such vertices if the two binary strings differ in at most 2 bits. (for example for $d=2$ this is $K_4$. for $d=3$ this is the cube with all the face diagonals)
Is this graph and its spectrum known? If something is known about this graph's spectrum then cans someone kindly give the references?
This graph is usually known as the halved $n$-cube. It is discussed on page 264 of Brouwer, Cohen and Neumaier "Distance-Regular Graphs". From there we see that the eigenvalues are \[ \theta_j = \frac12(n-2j)^2-\frac12n \] with respective multiplicities $\binom{n}{j}$ when $2j<n$; if $n=2d$ then $\theta_d$ has multiplicity $\frac12\binom{n}{d}$.
I outline one way to get the eigenvalues. Let $A_1$ denote the adjacency matrix of the $n$-cube and let $A_2$ be the adjacency matrix of the graph with the same vertices as the $n$-cube, but with two vertices adjacent if and only if they are at distance two in the $n$-cube. (This graph has two components, each of which is a halved $n$-cube.) Then we have \[ A_1^2 = nI+2A_2 \] and therefore $A_2=\frac12(A_1^2-nI)$. The eigenvalues of the $n$-cube are the integers $n-2r$ for $r=0,\ldots,n$, so we get the eigenvalues of the halved $n$-cube s above.