Mathematics and Statistics Faculty PublicationsCopyright (c) 2015 Utah State University All rights reserved.
http://digitalcommons.usu.edu/mathsci_facpub
Recent documents in Mathematics and Statistics Faculty Publicationsen-usWed, 04 Feb 2015 01:47:05 PST3600Linear Operators That Preserve Graphical Properties of Matrices: Isolation Numbers
http://digitalcommons.usu.edu/mathsci_facpub/198
http://digitalcommons.usu.edu/mathsci_facpub/198Mon, 02 Feb 2015 12:51:02 PST
Let A be a Boolean {0, 1} matrix. The isolation number of A is the maximum number of ones in A such that no two are in any row or any column (that is they are independent), and no two are in a 2 × 2 submatrix of all ones. The isolation number of A is a lower bound on the Boolean rank of A. A linear operator on the set of m × n Boolean matrices is a mapping which is additive and maps the zero matrix, O, to itself. A mapping strongly preserves a set, S, if it maps the set S into the set S and the complement of the set S into the complement of the set S. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that T is a Boolean linear operator that strongly preserves isolation number k for any 1 ⩽ k ⩽ min{m, n} if and only if there are fixed permutation matrices P and Q such that for X∈m,n(𝔹)T(X)=PXQ or, m = n and T (X) = PX t Q where X t is the transpose of X.
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LeRoy B. Beasley et al.Stability of traveling-wave solutions for a Schrodinger system with power-type nonlinearities
http://digitalcommons.usu.edu/mathsci_facpub/197
http://digitalcommons.usu.edu/mathsci_facpub/197Tue, 25 Nov 2014 14:44:29 PST
In this article, we consider the Schrodinger system with powertype nonlinearities,(Formula presented) where j = 1,...,m, uj are complex-valued functions of (x, t) 2 RN+1, a, b are real numbers. It is shown that when b > 0, and a + (m - 1)b > 0, for a certain range of p, traveling-wave solutions of this system exist, and are orbitally stable.
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Nghiem Nguyen et al.Self-Assessment: Aiding Awareness of Achievement
http://digitalcommons.usu.edu/mathsci_facpub/195
http://digitalcommons.usu.edu/mathsci_facpub/195Mon, 18 Mar 2013 15:03:22 PDTDavid E. BrownThe Development and Evaluation of a Program for Improving and
Assessing the Teaching of Mathematics and Statistics
http://digitalcommons.usu.edu/mathsci_facpub/196
http://digitalcommons.usu.edu/mathsci_facpub/196Mon, 18 Mar 2013 15:03:22 PDTDavid E. Brown et al.Assessing Proofs with Rubrics: The RVF Method
http://digitalcommons.usu.edu/mathsci_facpub/194
http://digitalcommons.usu.edu/mathsci_facpub/194Mon, 18 Mar 2013 15:03:21 PDTDavid E. BrownProbe Interval Orders
http://digitalcommons.usu.edu/mathsci_facpub/193
http://digitalcommons.usu.edu/mathsci_facpub/193Mon, 18 Mar 2013 15:03:20 PDTDavid E. Brown et al.Variations on Interval Graphs
http://digitalcommons.usu.edu/mathsci_facpub/192
http://digitalcommons.usu.edu/mathsci_facpub/192Mon, 18 Mar 2013 15:03:19 PDTDavid E. Brown et al.Characterizations of Interval Bigraphs and Unit Interval
Bigraphs
http://digitalcommons.usu.edu/mathsci_facpub/191
http://digitalcommons.usu.edu/mathsci_facpub/191Mon, 18 Mar 2013 15:03:18 PDTDavid E. Brown et al.Interval k-graphs
http://digitalcommons.usu.edu/mathsci_facpub/190
http://digitalcommons.usu.edu/mathsci_facpub/190Mon, 18 Mar 2013 15:03:17 PDTDavid E. Brown et al.Relationships Among Classes of Interval Bigraphs, (0,1)-matrices, and
Circular Arc Graphs
http://digitalcommons.usu.edu/mathsci_facpub/189
http://digitalcommons.usu.edu/mathsci_facpub/189Mon, 18 Mar 2013 15:03:16 PDTDavid E. Brown et al.On Cycle and Bi-cycle Extendability in Chordal and Chordal Bipartite Graphs
http://digitalcommons.usu.edu/mathsci_facpub/188
http://digitalcommons.usu.edu/mathsci_facpub/188Mon, 18 Mar 2013 15:03:14 PDTLeRoy B. Beasley et al.Bipartite Probe Interval Graphs, Interval Point Bigraphs, and Circular ArcGraphs
http://digitalcommons.usu.edu/mathsci_facpub/187
http://digitalcommons.usu.edu/mathsci_facpub/187Mon, 18 Mar 2013 15:03:13 PDTDavid E. Brown et al.A Characterization of Cycle-free Unit Probe Interval Graphs
http://digitalcommons.usu.edu/mathsci_facpub/185
http://digitalcommons.usu.edu/mathsci_facpub/185Mon, 18 Mar 2013 15:03:12 PDT
A graph is a probe interval graph (PIG) if its vertices can be partitioned into probes and nonprobes with an interval assigned to each vertex so that vertices are adjacent if and only if their corresponding intervals overlap and at least one of them is a probe. PIGs are a generalization of interval graphs introduced by Zhang for an application concerning the physical mapping of DNA in the human genome project. PIGs have been characterized in the cycle-free case by Sheng, and other miscellaneous results are given by McMorris, Wang, and Zhang. Johnson and Spinrad give a polynomial time recognition algorithm for when the partition of vertices into probes and nonprobes is given. The complexity for the general recognition problem is not known. Here, we restrict attention to the case where all intervals have the same length, that is, we study the unit probe interval graphs and characterize the cycle-free graphs that are unit probe interval graphs via a list of forbidden induced subgraphs.
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David E. Brown et al.Interval Tournaments
http://digitalcommons.usu.edu/mathsci_facpub/186
http://digitalcommons.usu.edu/mathsci_facpub/186Mon, 18 Mar 2013 15:03:12 PDT
A tournament is an orientation of a complete graph. A directed graph is an interval digraph if for each vertex v there corresponds an ordered pair of intervals (Sv, Tv) such that u → v if and only if Su ∩ Tv ≠ ∅. A bipartite graph is an interval bigraph if to each vertex there corresponds an interval such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex belongs to a different partite set. We use the equivalence of the models for interval digraphs and interval bigraphs to characterize tournaments that are interval digraphs via forbidden subtournaments and prove that a tournament on n vertices is an interval digraph if and only if it has a transitive (n − 1)-subtournament. We also characterize the obstructions to the existence of a transitive subtournament of order n − 1 in a tournament of order n.
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David E. Brown et al.Extending Partial Tournanents
http://digitalcommons.usu.edu/mathsci_facpub/184
http://digitalcommons.usu.edu/mathsci_facpub/184Mon, 18 Mar 2013 15:03:11 PDT
Let A be a (0,1,∗)-matrix with main diagonal all 0’s and such that if ai,j=1 or ∗ then aj,i=∗ or 0. Under what conditions on the row sums, and or column sums, of A is it possible to change the ∗’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments. The result we give is best possible in the sense that no relaxation of any condition will always yield a matrix that can be so extended.
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LeRoy B. Beasley et al.Preserving Regular Tournaments and Term Rank-1
http://digitalcommons.usu.edu/mathsci_facpub/183
http://digitalcommons.usu.edu/mathsci_facpub/183Mon, 18 Mar 2013 15:03:10 PDT
We investigate linear operators which map certain types of tournaments to themselves. To this end we also characterize term rank-1 preservers on the set of matrices whose associated digraphs are simple loopless directed graphs, and find that this set of operators is more diverse than might be expected.
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LeRoy B. Beasley et al.Embedding Tournaments
http://digitalcommons.usu.edu/mathsci_facpub/182
http://digitalcommons.usu.edu/mathsci_facpub/182Mon, 18 Mar 2013 15:03:09 PDTLeRoy B. Beasley et al.Cycle Extendability in Graphs and Digraphs
http://digitalcommons.usu.edu/mathsci_facpub/181
http://digitalcommons.usu.edu/mathsci_facpub/181Mon, 18 Mar 2013 15:03:08 PDT
In 1990, Hendry conjectured that all chordal Hamiltonian graphs are cycle extendable, that is, the vertices of each non-Hamiltonian cycle are contained in a cycle of length one greater. Let A be a symmetric (0,1)-matrix with zero main diagonal such that A is the adjacency matrix of a chordal Hamiltonian graph. Hendry’s conjecture in this case is that every k×k principle submatrix of A that dominates a full cycle permutation k×k matrix is a principle submatrix of a (k+1)×(k+1) principle submatrix of A that dominates a (k+1)×(k+1) full cycle permutation matrix. This article generalizes the concept of cycle-extendability to S-extendable; that is, with S⊆{1,2,…,n} and G a graph on n vertices, G is S-extendable if the vertices of every non-Hamiltonian cycle are contained in a cycle length i greater, where i∈S. We investigate this concept in directed graphs and in particular tournaments, i.e., anti-symmetric matrices with zero main diagonal.
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LeRoy B. Beasley et al.Linear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance and Bipartite Probe Interval Graphs
http://digitalcommons.usu.edu/mathsci_facpub/180
http://digitalcommons.usu.edu/mathsci_facpub/180Mon, 18 Mar 2013 15:03:07 PDT
A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V,E) is a tolerance graph if each vertex v ∈V can be associated to an interval Iv of the real line and a positive real number tv such that uv ∈E if and only if |Iu ∩Iv| ≥ min (tu,tv). In this paper we present O(|V| + |E|) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms.
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David E. Brown et al.Boolean Rank of Upset Tournament Matrices
http://digitalcommons.usu.edu/mathsci_facpub/179
http://digitalcommons.usu.edu/mathsci_facpub/179Mon, 18 Mar 2013 15:03:06 PDT
The Boolean rank of an m×n(0,1)-matrix M is the minimum k for which matrices A and B exist with M=AB, A is m×k, B is k×n, and Boolean arithmetic is used. The intersection number of a directed graph D is the minimum cardinality of a finite set S for which each vertex v of D can be represented by an ordered pair (Sv,Tv) of subsets of S such that there is an arc from vertex u to vertex v in D if and only if Su∩Tv≠Ø. The intersection number of a digraph is equal to the Boolean rank of its adjacency matrix. Using this fact, we show that the intersection number of an upset tournament, equivalently, the Boolean rank of its adjacency matrix, is equal to the number of maximal subpaths of certain types in its upset path.
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David E. Brown et al.