Geir Dahl

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 $L_G$ of the graph $G$. This problem corresponds to the minimization of a quadratic form associated with $L_G$, under certain constraints involving the $\ell_2$-norm. We introduce and investigate a similar problem, but using the $\ell_1$-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 $\ell_{\infty}$-norm version of the problem.

We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.