20 research outputs found
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
vertices with distinct eigenvalues is upper-bounded by .
Computations on trees up to 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
Unicyclic graphs and the inertia of the distance squared matrix
A result of Bapat and Sivasubramanian gives the inertia of the distance
squared matrix of a tree. We develop general tools on how pendant vertices and
degree 2 vertices affect the inertia of the distance squared matrix and use
these to give an alternative proof of this result. We further use these tools
to extend this result to certain families of unicyclic graphs, and we explore
how far these results can be extended
Graphs with few trivial characteristic ideals
We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1