Combinatorial Fiedler theory and graph partition

Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a(G) of the Laplacian matrix LG of the graph G. This problem corresponds to the minimization of a quadratic form associated with LG, under certain constraints involving the ℓ2-norm. We introduce and investigate a similar problem, but using the ℓ1-norm to measure distances. This leads to a new parameter b(G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b(G) for trees. We also comment on an ℓ∞-norm version of the problem.