Assign new point to a class using spectral clustering
I think you do not, since I was able to find several papers that proposed algorithms for the same:


Incremental spectral clustering by efficiently updating the eigen-system
Incremental kernel spectral clustering for online learning of non-stationary data
A Fast Incremental Spectral Clustering for Large Data Sets


Such algorithms are called sequential, incremental, streaming, or online. Armed with this knowledge, you can find more papers on your own.