# kMeans on graph Laplacian to cluster nodes based on their distance

I have a connected weighted graph and I want to use kMeans to cluster the points based on their distance (smaller distances indicate that the nodes are more likely to be in the same cluster).

I computed the Laplacian of the graph and chose the eigenvectors that have corresponding nonzero eigenvalues. I then performed the kMeans in this embedding space represented by the chosen eigenvectors.

In the following example, I want to create 2 clusters and I chose nodes $$1$$ and $$3$$ to be the initial centroids. My problem is that the resulting clustering puts nodes $$2, 3, 4$$ in one cluster and the node $$1$$ in another one. The desired clustering is $$1, 2, 4$$ in one cluster and $$3$$ in the other one.

Does it make sense to use such an approach to cluster the nodes of a graph? Or how can it be that the resulting clustering does not minimize the distances between the points in one cluster? I am using sklearn.cluster.k-means.