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.
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: