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Aida Abiad (Eindhoven University of Techonolgy)5/19/26, 11:00 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
A unified framework of the Expander Mixing Lemma for irregular graphs using adjacency eigenvalues will be presented, as well as several new versions of it. We will also show some of its applications in graph theory, which include spectral bounds on the zero forcing number of a graph. To derive our results we use a new application of weight partitions of graphs, where the Perron eigenvector...
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Shaun Fallat (University of Regina)5/19/26, 11:25 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
An inverse eigenvalue problem for a graph (IEP-G) asks a fundamental question: What are the possible spectra for (symmetric) real matrices fitting a given graph? Many have worked on several aspects of the IEP-G with exciting advances and variations appearing over the past forty years. Here, the focus will be on weighted Laplacian matrices associated with a graph. Such matrices are permanently...
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Kevin Vander Meulen (Redeemer University)5/19/26, 2:00 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
A matrix $A$ has the non-symmetric strong spectral property (nSSP) if $X=O$ is the only matrix which satisfies $A\circ X=O$ and $AX^T=X^TA$. This property comes with implications for eigenvalue properties of sign patterns, including a bifurcation lemma and superpattern lemma. We describe some classes of sign patterns for which every matrix with the sign pattern will have the nSSP. We also...
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Jephian C.-H. Lin (National Yang Ming Chiao Tung University)5/19/26, 2:25 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
A sign pattern is a matrix whose entries are in $\{+, -, 0\}$, while its quantitative class is the set of real matrices whose entries match the corresponding signs. A sign pattern is said to be spectrally arbitrary if its quantitative class contains matrices demonstrating all possible monic real polynomials as the characteristic polynomials. Historically, there are the Jacobian method, the...
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H Tracy Hall (none)5/19/26, 2:50 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
The Inverse Eigenvalue Problem for a Graph (IEP-G) asks for the possible spectra of a real symmetric matrix knowing only which off-diagonal entries are non-zero, as described by a graph $G$. Three matrix properties, collectively called the “strong properties”, have become prominent in the study of this problem, due in part to their good behavior with respect to edge deletion and contraction...
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Carolyn Reinhart (Swarthmore College)5/20/26, 10:45 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
Zero forcing is a graph coloring process in which a set of initially blue vertices force the remaining vertices in the graph to be colored blue after repeated applications of a color change rule. Leaky forcing is a fault-tolerant variant of zero forcing in which some set of $\ell$ vertices, called leaks, are forbidden from forcing. The $\ell$-leaky forcing number is the size of the smallest...
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Bonnie Jacob (Rochester Institute of Technology)5/20/26, 11:10 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
In this talk, we introduce a new parameter, the orientable forcing number of an undirected graph $G$, which is the maximum zero forcing number among all oriented graphs that have $G$ as their underlying undirected graph. We establish some properties of the orientable forcing number, including extreme values, and discuss how the parameter relates to matrices.
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Mary Flagg (University of St. Thomas)5/20/26, 11:35 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
Given a simple graph $G$, $\mathcal{S}(G)$ is the set of real symmetric matrices indexed by the vertices in $G$ and with off-diagonal zeros corresponding to non-edges in $G$. The problem of finding the maximum nullity of a matrix in $\mathcal{S}(G)$ has been extensively studied. We consider the maximum nullity of a matrix $A$ and its principal submatrix $A(i)$ corresponding to deleting the...
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Caleb Cheung (University of Wyoming)5/21/26, 11:00 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
Let $A$ be an $m \times n$ matrix. The spoiler space of $A$ is the set of all $m\times n$ matrices $X$ such that, $XA^\top$ is symmetric, $A^\top X$ is symmetric, and $X\circ A = O$, where "$\circ$" denotes the entrywise Schur product. If the spoiler space of $A$ contains only the 0 matrix, we say that $A$ has the Strong Singular Value Property (SSVP). The SSVP gives us access to a rich...
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Himanshu Gupta (University of Regina)5/21/26, 11:25 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
Symplectic geometry appears in many areas of mathematics, physics, and applications, and naturally gives rise to interesting matrix families and properties. Symplectic eigenvalues extend the classical notion of eigenvalues to the symplectic setting and are guaranteed to exist for positive definite matrices by Williamson's theorem. We introduce the inverse symplectic eigenvalue problem for...
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Dr Minerva Catral (Xavier University)5/21/26, 11:50 AMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
For vertex-labelled graphs $G$ and $H$ on $n\geq 1$ vertices, we consider matrices of the form $C(A,B) = \left[\begin{array}{c|c} A&B\\ \hline I&O\\\end{array}\right]\in\mathbb{R}^{2n\times 2n}$ where $A,B\in\mathbb{R}^{n\times n}$ are a pair of real symmetric matrices with nonzero patterns determined by the edges of the graph pair $G, H$. We denote the set of all such matrices by...
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Mark Hunnell (Winston-Salem State University)5/21/26, 2:25 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
The minimum rank of a graph $G$ of order $n$ is the smallest possible rank over all real symmetric $n\times n$ matrices $A$ whose $(i,j)$th entry, for $i\neq j$, is nonzero whenever $ij$ is an edge of $G$ and zero otherwise. We discuss some refinements of techniques currently in the literature to determine the minimum rank of a graph, some new tools to bound this value, and an approach for...
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Mark Kempton (Brigham Young University)5/21/26, 2:50 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
One of the most challenging aspects of graph inverse eigenvalue problems is knowing when a graph admits a matrix with few distinct eigenvalues. We will discuss results where we construct matrices with only two distinct eigenvalues corresponding to graphs that arise from various kinds of product structures. We will discuss also some directions for future work along these lines.
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Dallin Seyfried (Brigham Young University)5/21/26, 3:15 PMThe Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
An isospectral reduction is a method of shrinking a large matrix into a smaller matrix while preserving properties of the original's spectrum. The inverse, an isospectral unfolding, takes a matrix of an isospectral reduction and expands it into a larger matrix that has that reduction. We present a system of nonlinear equations forming the foundation of general isospectral unfolding. Graphs...
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Dr Hein Van der Holst (Georgia State University)The Inverse Eigenvalue Problem of a Graph and Zero ForcingMinisymposium Talk
For a multi-digraph $D$ with vertex-set $V=\{1,\ldots,n\}$ and arc-set $A$, let $Q(D)$ be the set of all real $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}\not=0$ if $i\not=j$ and there is a single arc from $i$ to $j$, $a_{i,j}\in \mathbb{R}$ if $i\not=j$ and there are multiple arcs from $i$ to $j$, $a_{i,j}=0$ if $i\not=j$ and there is no arc from $i$ to $j$, $a_{i,i}\not=0$ if there is no...
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