Proofs obtained only through spectral graph theory I have an increasing interest in spectral graph theory, which I find fascinating, and I've started collecting a few documents that I have yet to read more thoroughly than what I so far have.

However, I'm curious about a statement that popped up in several sources (for instance over there), which says in essence that some results in graph theory have been proved using spectrum-based techniques only, and that so far, no proof that bypasses those techniques is known. 

Unless I skipped that, I don't recall seeing such an example in the literature I've read so far. Do any of you know of examples of such results?
The Hoffman–Singleton theorem.