Michael S. Cavers

ON DETERMINING MINIMAL SPECTRALLY ARBITRARY PATTERNS

A new family of minimal spectrally arbitrary patterns is presented which allow for arbitrary spectrum by using the Nilpotent-Jacobian method introduced in (J.H. Drew, C.R. Johnson, D.D. Olesky, and P. van den Driessche. Spectrally arbitrary patterns.Lin. Alg. and Appl. 308:121- 137, 2000). The novel approach here is the use of the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns.

INERTIALLY ARBITRARY NONZERO PATTERNS OF ORDER 4

Inertially arbitrary nonzero patterns of order at most 4 are characterized. Some of these patterns are demonstrated to be inertially arbitrary but not spectrally arbitrary. The order 4 sign patterns which are inertially arbitrary and have a nonzero pattern that is not spectrally arbitrary are also described. There exists an irreducible nonzero pattern which is inertially arbitrary but has no signing that is inertially arbitrary. In fact, up to equivalence, this pattern is unique among the irreducible order 4 patterns with this property.

Allow problems concerning spectral properties of patterns

Let S � {0,+, ,+0, 0,�,#} be a set of symbols, where + (resp., , +0 and 0) denotes a positive (resp., negative, nonnegative and nonpositive) real number, and � (resp., #) denotes a nonzero (resp., arbitrary) real number. An S-pattern is a matrix with entries in S. In particular, a {0,+,} -pattern is a sign pattern and a {0,�}-pattern is a zero-nonzero pattern. This paper extends the following problems concerning spectral properties of sign patterns and zero-nonzero patterns to S-patterns: spectrally arbitrary patterns; inertially arbitrary patterns; refined inertially arbitrary patterns; potentially nilpotent patterns; potentially stable patterns; and potentially purely imaginary patterns. Relationships between these classes of S-patterns are given and techniques that appear in the literature are extended. Some interesting examples and properties of patterns when # belongs to the symbol set are highlighted. For example, it is shown that there is a {0,+,#}-pattern of order n that is spectrally arbitrary with exactly 2n 1 nonzero entries. Finally, a modified version of the nilpotent-Jacobian method is presented that can be used to show a pattern is inertially arbitrary. 1. Introduction: Definitions, background and motivation. 1.1. Definitions. Throughout this paper, we assume all matrices are square and use the notation S to denote the set of symbols S = f0;+;−;+0;−0;�;#g, where + (resp., −) represents a positive (resp., negative) real number, +0 (resp., −0) represents a nonnegative (resp., nonpositive) real number, and � (resp., #) represents a nonzero (resp., arbitrary) real number. For a symbol set SS, an S-pattern is a matrix with entries in S. In particular, a f0;+;−g-pattern is a sign pattern, a f0;�g-pattern is a zero-nonzero pattern, a f+;−g-pattern is a full sign pattern, a f0;+g-pattern is a nonnegative sign pattern, a f+g-pattern is a positive sign pattern, and a f0;+;−;#g- pattern is a generalized sign pattern. We use the term pattern when statements hold for all S-patterns with SS.

Techniques for identifying inertially arbitrary patterns

Two techniques used to show a matrix pattern is spectrally arbitrary are the nilpotent-Jacobian method and more recently the nilpotent-centralizer method. This paper presents generalizations of both techniques, which are then used to show that certain non-spectrally-arbitrary patterns are inertially arbitrary. A flaw in a method used in three previous publications on inertially arbitrary patterns is discussed. By using the techniques developed here, it is shown that all of the patterns in the three papers affected by the flaw are nevertheless inertially arbitrary.

Minimum Number of Distinct Eigenvalues of Graphs

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

Spatio-Temporal Complex Markov Chain (SCMC) Model Using Directed Graphs: Earthquake Sequencing

We construct a directed graph to represent a Markov chain of global earthquake sequences and analyze the statistics of transition probabilities linked to earthquake zones. We use a simplified plate boundary template for earthquake zonation. We generalize this Markov chain of earthquake sequences by including the recurrent events in space and time for each event in the record-breaking sense. The record-breaking recurrent events provide the basis for redefining the weights for the state-to-state transition probabilities. We use a distance-dependent look-up array for each zone to assign the distance-dependent weights for the recurring events. We present here details of the method and the preliminary results on the structure and properties of the directed graphs corresponding to a Markov chain model without and with the inclusion of record-breaking events. The underlying directed graph provides the framework for earthquake sequencing. We examine the properties of the directed graph without and with the inclusion of recurrences. We consider the present method easily expandable for forecasting work as catalogues are routinely updated with seismic events and, also, widely applicable to a study of both the regional and global seismicity. We demonstrate the applicability of the directed graph approach to forecasting using some of the properties of graphs that represent the Markov chain.

On reducible matrix patterns

A tool to study the inertias of reducible nonzero (resp. sign) patterns is presented. Sumsets are used to obtain a list of inertias attainable by the pattern 𝒜 ⊕ ℬ dependent upon inertias attainable by patterns 𝒜 and ℬ. It is shown that if ℬ is a pattern of order n, and 𝒜 is an inertially arbitrary pattern of order at least 2(n − 1), then 𝒜 ⊕ ℬ is inertially arbitrary if and only if ℬ allows the inertias (0, 0, n), (0, n, 0) and (n, 0, 0). We illustrate how to construct other reducible inertially (resp. spectrally) arbitrary patterns from an inertially (resp. spectrally) arbitrary pattern 𝒜 ⊕ ℬ, by replacing 𝒜 with an inertially (resp. spectrally) arbitrary pattern 𝒮. We identify reducible inertially (resp. spectrally) arbitrary patterns of the smallest orders that contain some irreducible components that are not inertially (resp. spectrally) arbitrary. It is shown there exist nonzero (resp. sign) patterns 𝒜 and ℬ of orders 4 and 5 (resp. 4 and 4) such that both 𝒜 and ℬ are non-inertially-arbitrary, and 𝒜 ⊕ ℬ is inertially arbitrary.

Distinguishing chromatic numbers of complements of Cartesian products of complete graphs

Abstract The distinguishing chromatic number of a graph, G , is the minimum number of colours required to properly colour the vertices of G so that the only automorphism of G that preserves colours is the identity. There are many classes of graphs for which the distinguishing chromatic number has been studied, including Cartesian products of complete graphs (Jerebic and Klavžar, 2010). In this paper we determine the distinguishing chromatic number of the complement of the Cartesian product of complete graphs, providing an interesting class of graphs, some of which have distinguishing chromatic number equal to the chromatic number, and others for which the difference between the distinguishing chromatic number and chromatic number can be arbitrarily large.

Confidence Weighting Procedures for Multiple-Choice Tests

Multiple-choice tests are extensively used in the testing of mathematics and statistics in undergraduate courses. This paper discusses a confidence weighting model of multiple choice testing called the student-weighted model. In this model, students are asked to indicate an answer choice and their certainty of its correctness. This method was implemented in two first year Calculus courses at the University of Calgary in 2014 and 2015. The results of this implementation are discussed here.

Clique partitions of distance multigraphs

We consider the minimum number of cliques needed to partition the edge set of D(G), the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a fractional analogue of this parameter to find lower bounds for the distance multigraphs of various classes of graphs. Some of the bounds are shown to be exact.