Why spectral clustering results in disjointed cluster?
First, please note that spectral clustering is very sensitive to the affinity kernel. With the standard RBF kernel, my experience is that spectral clustering often isolates outliers (in the spectral space), leaving clusters with numerous observations which can be separated by great distances. This is a major difference with direct k-means: there is no notion of distance anymore, it's just about connectivity.

In the case shown in your example, the results are "strange" because of the low dimensionality: 1D is typically a case where spectral clustering does not provide "logical" results. In addition, you are quite unlucky on several aspects:


The numbers of clusters (4): try it out with different numbers of clusters, and you will find more trivial results
The positions of observations: move #0 to the left and you may find results that make more sense
The low number of observations: #6 is not considered as an outlier, due to its vicinity to the #5, #8, #9 triplet. Indeed, the more points there are in an area, the largest its influence is. This is also how #4 and #7 can be considered part of the (#5, #6, #8, #9) cluster, rather than the (#2, #3) cluster. Try removing #9 and see how the results suddenly make sense.
The chosen kernel (not tried on my side, but it is likely that the results may vary a lot if you increase the gamma value or change the kernel shape)


TL;DR: your case is an example of how spectral clustering can go wrong with low number of observations and dimensions.