9:00-9:30
Shaun Fallat
Title: Bryan's relentless desire to find a "generic" matrix
Abstract: Bryan's contributions to combinatorial matrix theory are significant and numerous. Just as essential, in my view, is his "door's always open" approach to other researchers, both new and old, wanting to learn about his existing work, new thoughts or just have the opportunity to join his many research ventures. This certainly applies to me, as I am indebted to Bryan for his mentor-ship and guidance over the past 20 years of our partnership! A considerable portion of Bryan's recent aim has been to utilize the notion of a "generic" matrix to resolve existing open problems and advance the discipline in many exciting directions. I will survey some of these interesting topics, many of which have been thoughtfully engineered by Bryan and his phenomenal vision. HAPPY BIRTHDAY Bryan!
9:30-10:00
Sudipta Mallik
Title: Fantastic incidence matrices and how to invert them
Abstract: Consider a simple graph G on n vertices 1, 2, . . . , n with m edges e_1, e_2, . . . , e_m. The vertex-edge incidence matrix of G, denoted by M, is the n X m matrix whose (i, j)-entry is 1 if vertex i is incident with edge e_j and 0 otherwise. We will investigate a generalized inverse of M called the Moore-Penrose inverse of M, denoted by M+. A combinatorial formula of the entries of M+ will be presented when G is a tree, unicyclic graph, and wheel graph. The same will be discussed for signless Laplacian matrix Q=MM^T. We will present open problems some of which are accessible for undergraduate students.
10:30-11:00
Kevin vanderMeulen
Title: Matrix companions: combinatorial insights
Abstract: The Frobenius companion matrix is well-known and can be described as a matrix template for obtaining a particular characteristic polynomial. Combinatorial analysis has led to the discovery of other companion matrices, that is, other matrix templates for obtaining a prescribed characteristic polynomial. The journey to discovery will be presented, along with some open problems.
11:00-11:30
Steve Kirkland
Title: Shader's Sharp Observation
Abstract: In some collaborative work with myself and Michael Neumann, Bryan Shader made a key observation that provides a combinatorial perspective on the group inverse of the Laplacian matrix of a graph. In this talk I will describe Bryan's original insight, then discuss how it furnishes a combinatorial interpretation of Kemeny's constant for Markov chains, and informs a control strategy in the context of a model for cholera.
1:30-2:00
Colin Garnett
Title: Edge Turncoat Graphs: The good, the bad and the ugly
Abstract: Cops and robbers is a vertex pursuit game played on a simple graph. The win/loss state for each graph is known and well studied. We ask the question, when does the removal or addition of any edge change this win/loss state? We focus on the case when adding an edge changes the state from robber-win to cop-win. These are called maximally robber-win graphs, while we have not found a characterization of such graphs, we have come up with a variety of examples.
2:30-3:00
Bryan Curtis
Title: New results for Sign Patterns of Orthogonal Matrices
Abstract: A sign pattern is a combinatorial tool that describes when the entries of a real matrix are positive, negative or zero. Classifying the sign patterns of orthogonal matrices is a long-standing open problem. The strong inner product property is a useful tool for constructing examples of sign patterns of orthogonal matrices. New results related to the strong inner product property are given. We also discuss asymptotic results for sign patterns of row orthogonal matrices.
3:00-3:30
Michael Adams
Title: Lining Up Electrons
Abstract: A few years ago a colleague in physics asked me to consider a combinatorial problem that had come up in his research on conductivity of metallic compounds. What he needed was a closed form expression that enumerated the number of ways to arrange electrons in a one-dimensional crystal that accounted for the effects of intersite Coulomb interactions as well as the spin states of the electrons. Using rational generating functions in several variables we have obtained some partial results, but a complete solution remains elusive. In this informal talk, I will discuss the methods that have yielded partial results and describe our current work.