A real n by n matrix B is a P_0^+ -matrix if for each k in {1, 2, . . . , n} every k by k princip... more A real n by n matrix B is a P_0^+ -matrix if for each k in {1, 2, . . . , n} every k by k principal minor of B is nonnegative, and at least one k by k principal minor is positive. A digraph D is said to have P_0^+-completion if every partial P_0^+-matrix specifying D can be completed to a P_0^+ -matrix. In this paper, we study the P_0^+-completion problem, give necessary conditions for a digraph to have P_0^+-completion, and single out those digraphs of order at most four that have P_0^+-completion.
Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining a... more Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white, then change the color of $v$ to black. A set $Z$ is a skew zero forcing set for $G$ if the application of the skew color change rule (as many times as necessary) will result in all the vertices in $G$ colored black. A set $Z$ is a minimum skew zero forcing set for $G$ if it is a skew zero forcing set for $G$ of least cardinality. The skew zero forcing number $\sZ (G)$ is the minimum of $|Z|$ over all skew zero forcing sets $Z$ for $G$. In this paper we discuss graphs that have extreme zero forcing number. We characterize complete multipartite graphs, and connected graphs $G$, with $\smr (\field, G) = 4$ ($\field$ an infinite field), in terms of $\sZ (G)$. We note relations between minimum skew zero forcing sets and matchings in some bipartite graph...
In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entr... more In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric P - matrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partialweakl y sign symmetric P0-matrix. Patterns of
In this paper the P0-matrix completion problem is considered. It is established that every asymme... more In this paper the P0-matrix completion problem is considered. It is established that every asymmetric partial P0-matrix has P0-completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P0-completion or not having P0-completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5h asP0-completion.
The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew ... more The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew symmetric matrices, whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. In this paper we study the problem of real minimum skew rank of powers and strict powers
A real n by n matrix B is a P_0^+ -matrix if for each k in {1, 2, . . . , n} every k by k princip... more A real n by n matrix B is a P_0^+ -matrix if for each k in {1, 2, . . . , n} every k by k principal minor of B is nonnegative, and at least one k by k principal minor is positive. A digraph D is said to have P_0^+-completion if every partial P_0^+-matrix specifying D can be completed to a P_0^+ -matrix. In this paper, we study the P_0^+-completion problem, give necessary conditions for a digraph to have P_0^+-completion, and single out those digraphs of order at most four that have P_0^+-completion.
Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining a... more Let $G$ be a graph, and $Z$ a subset of its vertices, which we color black, while the remaining are colored white. We define the skew color change rule as follows: if $u$ is a vertex of $G$, and exactly one of its neighbors $v$, is white, then change the color of $v$ to black. A set $Z$ is a skew zero forcing set for $G$ if the application of the skew color change rule (as many times as necessary) will result in all the vertices in $G$ colored black. A set $Z$ is a minimum skew zero forcing set for $G$ if it is a skew zero forcing set for $G$ of least cardinality. The skew zero forcing number $\sZ (G)$ is the minimum of $|Z|$ over all skew zero forcing sets $Z$ for $G$. In this paper we discuss graphs that have extreme zero forcing number. We characterize complete multipartite graphs, and connected graphs $G$, with $\smr (\field, G) = 4$ ($\field$ an infinite field), in terms of $\sZ (G)$. We note relations between minimum skew zero forcing sets and matchings in some bipartite graph...
In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entr... more In this paper it is shown that a partialsign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric P - matrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partialweakl y sign symmetric P0-matrix. Patterns of
In this paper the P0-matrix completion problem is considered. It is established that every asymme... more In this paper the P0-matrix completion problem is considered. It is established that every asymmetric partial P0-matrix has P0-completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P0-completion or not having P0-completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5h asP0-completion.
The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew ... more The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew symmetric matrices, whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. In this paper we study the problem of real minimum skew rank of powers and strict powers
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Papers by Luz Dealba