Shaun M. Fallat

Burning Signed Graphs

Shaun Fallat, Himanshu Gupta, Kamyar Khodamoradi +1

We introduce and analyze a new model of graph burning, in which two competing fires (coloured yellow and green) ignite vertices of a given signed graph and propagate along its edge…

math.CV2026

A proof of Borcea's quadratic variance conjecture

Ludovick Bouthat, Shaun Fallat

Let $p$ be a complex polynomial of degree $n \geq 2$ , and let $σ_{2} (p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every…

Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach

Jane Breen, Shaun Fallat, Johnna Parenteau

For any 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…

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.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…

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.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.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…

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.CO2025

The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture

Francesco Barioli, Shaun M. Fallat, Himanshu Gupta +1

Since the transformative workshop by the American Institute of Mathematics on the minimum rank of a graph, two longstanding open problems have captivated the community interested i…

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…

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…

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(\…

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…

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…

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…

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…

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★ 1 cited

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.CA2023★ 1 cited

Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace

Shaun M. Fallat, Prateek Kumar Vishwakarma

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is refer…

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.CO2023★ 1 cited

Bordering of Symmetric Matrices and an Application to the Minimum Number of Distinct Eigenvalues for the Join of Graphs

Aida Abiad, Shaun M. Fallat, Mark Kempton +5

An important facet of the inverse eigenvalue problem for graphs is to determine the minimum number of distinct eigenvalues of a particular graph. We resolve this question for the j…