David E. Brownhttps://works.bepress.com/david_brown/Recent works by David E. Brownen-usCopyright (c) 2019 All rights reserved.Sun, 25 Mar 2012 07:00:00 +00003600Full Isolation Number of Matrices: Some Extremal Resultshttps://works.bepress.com/david_brown/30/<p>A set of nonzero entries of a (0,1)-matrix is an isolated set if no two entries belong to the same row, no two entries belong to the same column, and no two entries belong to a submatrix of the form [1 1; 1 1]. The isolation number of a matrix is the maximum size over all isolated sets. The isolation number of a matrix is a well-known and well-used lower bound for the matrix's Boolean rank. We will discuss the isolation number of the adjacency matrix of various graphs and develop some extremal results for n x n matrices with isolation number n.</p>
David Tate et al.Sun, 25 Mar 2012 07:00:00 +0000https://works.bepress.com/david_brown/30/No Subject AreaForbidden Subgraph Characterization of Bipartite Unit Probe Interval Graphshttps://works.bepress.com/david_brown/12/David E. Brown et al.Sun, 01 Jan 2012 08:00:00 +0000https://works.bepress.com/david_brown/12/No Subject AreaIntersection Graph Theory Appliedhttps://works.bepress.com/david_brown/5/Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/david_brown/5/No Subject AreaCycle Extendability in Graphs and Digraphshttps://works.bepress.com/david_brown/27/<p>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.</p>
Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/david_brown/27/No Subject AreaBoolean Rank, Intersection Number, Dot-Product Dimensionhttps://works.bepress.com/david_brown/25/Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/david_brown/25/No Subject AreaCycle Extendability in Graphs, Bigraphs and Digraphshttps://works.bepress.com/david_brown/18/<p>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. In this talk, we discuss some preliminary results on a generalization of 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 present some results on tournaments, i.e., complete directed graphs, and some observations about cycle-extendability and S-extendability for non-directed graphs.</p>
Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/david_brown/18/No Subject AreaBoolean Rank of Upset Tournament Matriceshttps://works.bepress.com/david_brown/13/<p>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.</p>
Sat, 01 Jan 2011 08:00:00 +0000https://works.bepress.com/david_brown/13/No Subject AreaLinear Time Recognition Algorithms and Structure Theorems for Bipartite Tolerance and Bipartite Probe Interval Graphshttps://works.bepress.com/david_brown/15/<p>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.</p>
David E. Brown et al.Fri, 01 Jan 2010 08:00:00 +0000https://works.bepress.com/david_brown/15/No Subject AreaAssessing Proofs with Rubrics: The RVF Methodhttps://works.bepress.com/david_brown/23/<p>We present an easy-to-implement 3-axis rubric for the formative and summative assessment of open-ended solutions and proofs. The rubric was constructed for the use on the written work of students in a Discrete Mathematics class at a research-oriented university, with the following in mind: (1) To aid in the efficiency and consistency of assessment of proofs and open-ended solutions, with the possibility of being comfortably implemented by an undergraduate assistant; (2) To provide the simultaneous formative and summative assessment of the students’ written work. Thus, the questions we address are: How can we foster good technical writing skills in a way that improvement can be measured? How can large amounts of written work be processed and assessed so that summative and formative judgments are passed but without much time used by the instructor/professor/TA? The axes we use are labeled validity, readability, and fluency, corresponding to (respectively) correctness of calculations and deductions, the ease with which the solution or proof can be read, and the extent to which a student is able to use and communicate via the technical notions relevant to the problem or proof — for example appropriateness and correctness of notation. The rubric format is communicated to the students and discussed in class before any written work is assessed. The rubric has been implemented by professors and teaching assistants only after being trained in its use.</p>
David E. Brown et al.Fri, 01 Jan 2010 08:00:00 +0000https://works.bepress.com/david_brown/23/No Subject AreaCycle Extendabilityhttps://works.bepress.com/david_brown/19/Fri, 01 Jan 2010 08:00:00 +0000https://works.bepress.com/david_brown/19/No Subject Area