Uniqueness and minimal obstructions for tree-depth

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k-ranking of G exists. The graph G is k-critical if it has tree-depth k and every proper minor of G has smaller tree-depth. We establish partial results in support of two conjectures about the order and maximum degree of k-critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G, there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k-critical graphs are 1-unique, and we conjecture that the property holds for all k-critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k-critical graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure

Average mixing matrix of trees

We investigate the rank of the average mixing matrix of trees, with all eigenvalues distinct. The rank of the average mixing matrix of a tree on $n$ vertices with $n$ distinct eigenvalues is upper-bounded by $\frac{n}{2}$. Computations on trees up to $20$ vertices suggest that the rank attains this upper bound most of the times. We give an infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound. We also give a lower bound on the rank of the average mixing matrix of a tree.Comment: 18 pages, 2 figures, 3 table

Multi band Fermi surface in 1T-VSe2 and its implication for charge density wave phase

Here, our angle resolved photoemission spectroscopy experiment reveled that the surface band structure of the 1T-VSe2 host electronic states that was not predicted or probed before. Earlier claims to support charge density wave phase can be all explained in terms of these new findings. Its Fermi surface found to be not gaped at any point of the Brillouin zone and warping effect on the electronic structure, attributed to the lattice distortion previously, is due to the different dispersion of the multiple bands. Based on these new findings and interpretations, charge density wave induced modification on the electronic structure of 1T-VSe2 needs to be reconstructed in the future studies.Comment: