What is the spectrum of the Rado graph? Isn't this question self-explanatory?  There is a lot of literature about the Rado graph $R$ in various places.  This graph is also known as the "Random Graph" because a countable random graph is isomorphic to $R$ with probability 1.  There is also a lot of literature about spectra of graphs, finite and infinite.  The Rado graph is an exceptional object, and I would expect its spectrum to be interesting as well.  For that matter, its characteristic function should be interesting too.
 A: The infinite Rado graph could be specified as having vertices numbered $0,1,2,\cdots$ where there is an edge $(m,i)$ when the $i$th bit of the binary expansion of $m$ is a $1$. One could look at the induced graph on the  vertices $0,\cdots,n-1$ either for all $n$ or when $n$ is a power of $2$. As commented below, that is perhaps not the only choice. However it was an open ended question and I found that choice appealing. I had expected that things would be different right after a new power of $2$ compared to half way between two  such. Below is a plot of the eigenvalues up to $n=129.$
Some random observations about these $130$ cases:


*

*The number of distinct eigenvalues for n  from $0$ to $12$ are $1,2,3,4,5,6,7,7,9,9,9,8,9$

*Starting with $n=6$ There are $2k+3$ non-zero eigenvalues for $2^k \le n \lt 2^{k+1}.$ These are distinct with the exception of a double eigenvalue of$-2$ at $n=11.$ 

*There is an eigenvalue  of $0$ except for $n=1,3,4,5$. Hence, starting at $n=8$ it has multiplicity $n-2k-1$ for $k$  as above. That is; the multiplicity is $1$ at $n=8$ and then increases by $1$ when $n$ does, except that it drops by $2$ when $n$ is a power of $2.$

*The only non-zero values which occur for more than one $n$ (up to $n=127$) are


*

*$-2$ for $n=9,10,11,12,13$, 

*$+1$ for $n=1,4,10,11$ and 

*$-1$ for $n=3,4$


*The only integer eigenvalues not already mentioned are $+2$ for $n=35$ and$-4$ for $n=57$

