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Publications (34)

math.CO2023

The $q$ -Analogue of Zero Forcing for Certain Families of Graphs

Shaun Fallat, Neha Joshi, Roghayeh Maleki +6

Zero forcing is a combinatorial game played on a graph with the ultimate goal of changing the colour of all the vertices at minimal cost. Originally this game was conceived as a on…

math.CO2023

On the zero forcing number of the complement of graphs with forbidden subgraphs

Emelie Curl, Shaun Fallat, Ryan Moruzzi +2

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n - 3$ or $n - 1$ in one exceptional case, we consider the zero…

math.CO2014

Variants on the minimum rank problem: A survey II

Shaun Fallat, Leslie Hogben

The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$ th entry (for $i \neq = j$ ) is nonzero whenever…

math.SP2020

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Mohammad Adm, Shaun Fallat, Karen Meagher +3

Associated to a graph $G$ is a set $S (G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the gr…

math.CO2023

The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions

Sarah Allred, Emelie Curl, Shaun Fallat +4

The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently t…

math.RT2025

Linear Preservers of Real Matrix Classes Admitting a Real Logarithm

Shaun Fallat, Samir Mondal

In real Lie theory, matrices that admit a real logarithm reside in the identity component $GL_{n} (R)_{+}$ of the general linear group $GL_{n} (R)$ , wit…

math.CO2024

Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs

Shaun Fallat, Himanshu Gupta, Allen Herman +1

We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certa…

math.CO2018

Properties of a $q$ -analogue of zero forcing

Steve Butler, Craig Erickson, Shaun Fallat +6

Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation…

cs.LG2025

Distance-based Learning of Hypertrees

Shaun Fallat, Kamyar Khodamoradi, David Kirkpatrick +3

We study the problem of learning hypergraphs with shortest-path queries (SP-queries), and present the first provably optimal online algorithm for a broad and natural class of hyper…

math.CO2022

Sparsity of Graphs that Allow Two Distinct Eigenvalues

Wayne Barrett, Shaun Fallat, Veronika Furst +5

The parameter $q (G)$ of a graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$ . It is shown that the minimum number of edg…

math.CO2016

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

Wayne Barrett, Shaun Fallat, H. Tracy Hall +3

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for s…

math.CO2024

Inverse eigenvalue problem for Laplacian matrices of a graph

Shaun Fallat, Himanshu Gupta, Jephian C. -H. Lin

For a given graph $G$ , we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with $G$ . This new…

math.RA2026

Semigroup automorphisms of total positivity

Projesh Nath Choudhury, Shaun Fallat, Chi-Kwong Li

Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gant…

math.CO2020

The ErdÅs-Ko-Rado theorem for $2$ -intersecting families of perfect matchings

Shaun Fallat, Karen Meagher, Mahsa N. Shirazi

A perfect matching in the complete graph on $2 k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect mat…

math.CO2013

On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

Fatemeh Alinaghipour Taklimi, Shaun Fallat, Karen Meagher

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper…

math.CO2013

Minimum number of distinct eigenvalues of graphs

Bahman Ahmadi, Fatemeh Alinaghipour, Michael S. Cavers +3

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 $G$ , b…

math.CO2023

Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues

Wayne Barrett, Shaun Fallat, Veronika Furst +3

For an $n \times n$ matrix $A$ , let $q (A)$ be the number of distinct eigenvalues of $A$ . If $G$ is a connected graph on $n$ vertices, let $S (G)$ be the set of all real sy…

math.CO2026

The Strong Spectral Property and the Jacobian Method for Weighted Laplacian Matrices

Minerva Catral, Shaun Fallat, Himanshu Gupta +1

Strong matrix properties, roughly speaking, refer to generic conditions on a matrix such that its spectral perturbation and pattern perturbation interact nicely to cover a neighbor…

math.AC2021

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

Shaun Fallat, Charles R. Johnson, Alan D. Sokal

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t \ge r-2. W…

math.CO2023

The Spark of Symmetric Matrices Described by a Graph

Louis Deaett, Shaun Fallat, Veronika Furst +3

We investigate the sparsity of null vectors of real symmetric matrices whose off-diagonal pattern of zero and nonzero entries is described by the adjacencies of a graph. We use the…

math.CO2020

Complex Hadamard Diagonalisable Graphs

Ada Chan, Shaun Fallat, Steve Kirkland +3

In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex…

math.OC2026

Determinant Bounds for $(n - 1)$ -Locally Positive Semidefinite Matrices

Shaun Fallat, Samir Mondal, Hristo Sendov

In this framework, the extremal case corresponds to the tightest nontrivial relaxation in this hierarchy, in which every proper principal submatrix is constrained to be positive se…

math.CO2021

On the minimum number of distinct eigenvalues of a threshold graph

Shaun Fallat, Seyed Ahmad Mojallal

For a graph $G$ , we associate a family of real symmetric matrices, $S (G)$ , where for any $A \in S (G)$ , the location of the nonzero off-diagonal entries of $A$ are governed by the ad…

math.CO2014

On the Complexity of the Positive Semidefinite Zero Forcing Number

Shaun Fallat, Karen Meagher, Boting Yang

The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero f…

math.CO2024

Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues

Wayne Barrett, Shaun Fallat, Veronika Furst +3

The parameter $q (G)$ of an $n$ -vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$ . We show that all $G$ with $e(\…

math.CO2023

Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Shaun Fallat, Seyed Ahmad Mojallal

In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and a…

math.ST2016

Total positivity in Markov structures

Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi +3

We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence…

math.CO2026

Strictly Interlaced Spectral Data for the Weighted Matching Polynomial of a Graph

Shaun Fallat, Johnna Parenteau

Interlacing of the real roots of a weighted matching polynomial for a graph $G$ and that of a vertex-deleted subgraph is classical and well-known. In the context of strict interlac…

math.GM2026

Variations on Majorization of Vectors and Connections to Determinantal Inequalities

Shaun Fallat, Samir Mondal, Hristo Sendov

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matr…

math.CO2015

Compressed Cliques Graphs, Clique Coverings and Positive Zero Forcing

Shaun Fallat, Karen Meagher, Abolghasem Soltani +1

Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is w…

math.CO2016

Infection in Hypergraphs

Ryan Bergen, Shaun Fallat, Adam Gorr +5

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infecti…

math.GM2024

Results on the generalized numerical ranges in max algebra

Narges Haj Aboutalebi, Shaun Fallat, Aljosa Peperko +2

Let $n$ and $k$ be two positive integers with $k \leq n$ and $C$ an $n \times n$ matrix with nonnegative entries. In this paper, the rank- $k$ numerical range in the max algebra sett…

math.CO2023

On a relationship between the characteristic and matching polynomials of a uniform hypertree

Honghai Li, Li Su, Shaun Fallat

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$ -tree if, additionally, it is $r$ -uniform. Note that 2-trees are just ordinary trees. A cl…

math.CO2024

Sufficient conditions for total positivity, compounds, and Dodgson condensation

Shaun Fallat, Himanshu Gupta, Charles R. Johnson

A $n$ -by- $n$ matrix is called totally positive ( $T P$ ) if all its minors are positive and $T P_{k}$ if all of its $k$ -by- $k$ submatrices are $T P$ . For an arbitrary totally positive mat…

Publications (34)

The q-Analogue of Zero Forcing for Certain Families of Graphs

On the zero forcing number of the complement of graphs with forbidden subgraphs

Variants on the minimum rank problem: A survey II

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions

Linear Preservers of Real Matrix Classes Admitting a Real Logarithm

Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs

Properties of a q-analogue of zero forcing

Distance-based Learning of Hypertrees

Sparsity of Graphs that Allow Two Distinct Eigenvalues

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

Inverse eigenvalue problem for Laplacian matrices of a graph

Semigroup automorphisms of total positivity

The ErdÅs-Ko-Rado theorem for 2-intersecting families of perfect matchings

On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

Minimum number of distinct eigenvalues of graphs

Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues

The Strong Spectral Property and the Jacobian Method for Weighted Laplacian Matrices

Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples

The Spark of Symmetric Matrices Described by a Graph

Complex Hadamard Diagonalisable Graphs

Determinant Bounds for (n−1)-Locally Positive Semidefinite Matrices

On the minimum number of distinct eigenvalues of a threshold graph

On the Complexity of the Positive Semidefinite Zero Forcing Number

Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues

Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Total positivity in Markov structures

Strictly Interlaced Spectral Data for the Weighted Matching Polynomial of a Graph

Variations on Majorization of Vectors and Connections to Determinantal Inequalities

Compressed Cliques Graphs, Clique Coverings and Positive Zero Forcing

Infection in Hypergraphs

Results on the generalized numerical ranges in max algebra

On a relationship between the characteristic and matching polynomials of a uniform hypertree

Sufficient conditions for total positivity, compounds, and Dodgson condensation

The $q$ -Analogue of Zero Forcing for Certain Families of Graphs

Properties of a $q$ -analogue of zero forcing

The ErdÅs-Ko-Rado theorem for $2$ -intersecting families of perfect matchings

Determinant Bounds for $(n - 1)$ -Locally Positive Semidefinite Matrices