Minimum Ranks and Refined Inertias of Sign Pattern Matrices

A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. This thesis contains problems about refined inertias and minimum ranks of sign patterns. The refined inertia of a square real matrix $B$, denoted \ri(B), is the ordered $4$-tuple $(n_+(B), \ n_-(B), \ n_z(B), \ 2n_p(B))$, where $n_+(B)$ (resp., $n_-(B)$) is the number of eigenvalues of $B$ with positive (resp., negative) real part, $n_z(B)$ is the number of zero eigenvalues of $B$, and $2n_p(B)$ is the number of pure imaginary eigenvalues of $B$. The minimum rank (resp., rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of $\cal A$. First, we identify all minimal critical sets of inertias and refined inertias for full sign patterns of order 3. Then we characterize the star sign patterns of order $n\ge 5$ that require the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$, which is an important set for the onset of Hopf bifurcation in dynamical systems. Finally, we establish a direct connection between condensed $m \times n$ sign patterns and zero-nonzero patterns with minimum rank $r$ and $m$ point-$n$ hyperplane configurations in ${\mathbb R}^{r-1}$. Some results about the rational realizability of the minimum ranks of sign patterns or zero-nonzero patterns are obtained

Density and Chromatic Index, and Minimum Ranks of Sign Pattern Matrices

Given a (multi)graph, the density is defined by $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\}.$$ The {\bf chromatic index} $\chi\u27(G)$ of a graph $G$ is the minimum number of colors that required to color the edges of $G$ such that two adjacent edges receive different colors. It is known that $\chi\u27(G)\geq \Gamma(G)$. The {\bf cover index} $\xi(G)$ of $G$ is the greatest integer $k$ for which there is a coloring of $E$ with $k$ colors such that each vertex of $G$ is incident with at least one edge of each color. A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. In part 1, we will generally discuss the connections between the density and the chromatic index. In particular, the Goldberg-Seymour conjecture states that $\chi\u27(G)=\lceil\Gamma(G)\rceil$ if $\chi\u27(G)\u3e\Delta+1$, where $\Delta$ is the maximum degree of $G$. Some open problems are mentioned at the end of part 1. In particular, a dual conjecture to the Goldberg-Seymour conjecture on the cover index is discussed. A proof of the Goldberg-Seymour conjecture is given In part 2. In part 3, we will present a connection between the minimum ranks of sign pattern matrices and point-line configurations