John Sinkovic

THE INVERSE EIGENVALUE AND INERTIA PROBLEMS FOR MINIMUM RANK TWO GRAPHS

Let G be an undirected graph on n vertices and let S(G) be the set of all real sym- metric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are characterized; it is also noted that mr+(G) = α(G) for such graphs. This charac- terization solves the inverse inertia problem for graphs whose minimum rank is two. Furthermore, it is determined which diagonal entries are required to be zero, are required to be nonzero, or can be either for a rank minimizing matrix in S(G) when mr(G) = 2. Collectively, these results lead to a solution to the inverse eigenvalue problem for rank minimizing matrices for graphs whose minimum rank is two.

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.

Minimum ranks of sign patterns via sign vectors and duality

A {\it sign pattern matrix} is a matrix whose entries are from the set

Diagonal entry restrictions in minimum rank matrices

\{+,-, 0\}

A graph for which the inertia bound is not tight

. The minimum rank of a sign pattern matrix

On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs

A

Maximum nullity of outerplanar graphs and the path cover number

is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of

The minimum semidefinite rank of the complement of partial k-trees

A

The combinatorial inverse eigenvalue problem II: all cases for small graphs

. It is shown in this paper that for any

Minimum rank of outerplanar graphs

m \times n