Throughout this chapter, it was assumed that the underlying graph is undirected with non- negative edge weights. Considering the case when the edge weight can be negative or positive, discuss the effect on graph frequencies and their ordering based on the Laplacian quadratic form. What problems might arise in this case?

What will be an ideal response?

When there is no restriction on the signs of edge weights, some of the graph frequencies (the
eigenvalues of the graph Laplacian matrix) can be negative. Therefore, the frequency ordering
based on the Laplacian quadratic form will no longer result in zero eigenvalue as the lowest
frequency.