How to use spectral clustering to predict?
If the paper didn't elaborate on this, it must mean that they do 1). They look for the 10 closest neighbors of a new point and use majority voting to assign a cluster. On a side note, strange that they use an even number of neighbors, since it may introduce ties.

"What are the other ways to assign a(n) (existing) cluster to a new point?"

There are plenty of ways but it doesn't mean that they are all good! You could randomly assign points to clusters and use the size of the clusters as weight, but that isn't such a great method.

Generally, a nearest-neighbor approach is what is used. Now, it can be done in many ways:


KNN as I explained before (with any number of K)
Calculate the mean point of each cluster, and assign to the closest cluster
For each cluster, calculate the most distant point to the new point, then assign the cluster that has the "least far" distant point
And so on.