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Wei Gao5/18/26, 3:45 PMCombinatorial Matrix TheoryMinisymposium Talk
Let $G$ be a simple connected graph. A vertex-degree-based topological index is defined as $$TI_f(G) = \sum_{uv \in E(G)} f(d_u, d_v),$$ where $f(x, y)$ is a symmetric real function. In theoretical chemistry, these indices serve as essential numerical molecular descriptors in QSAR/QSPR models. In this work, we investigate the extremal properties of $TI_f + RTI_f$, defined as the sum of a...
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Nik Stopar (University of Ljubljana)5/18/26, 4:10 PMCombinatorial Matrix TheoryMinisymposium Talk
In this talk we demonstrate how a matrix algebra over a finite field can be completely described using combinatorial properties. The main tool that allows one to do this is the compressed zero-divisor graph of a ring, which describes pairs of matrices $A$ and $B$ such that $AB=0$. We list a set of $5$ combinatorial axioms that uniquely determine the compressed zero-divisor graph...
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Chris Hart (Georgia State University)5/18/26, 4:35 PMCombinatorial Matrix TheoryMinisymposium Talk
Let $A$ be an $m \times n$ real matrix. If the manifolds ${\widetilde{\cal M}_A}= \{ H^{-1} A G : G, H \text{ are nonsingular} \}$ and $Q(\text{sgn}(A))$ intersect transversally at $A,$ that is, the tangent spaces of ${\widetilde{\cal M}_A}$ and $Q(\text{sgn}(A))$ at $A$ sum to $ \mathbb R ^{m\times n},$ we say that $A$ has the rank-preserving transversality property (RPTP) and that $A$ is an...
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Johnna Parenteau5/18/26, 5:00 PMCombinatorial Matrix TheoryMinisymposium Talk
Given a simple graph, $G$, on $n$ vertices, let $S(G)$ be the set of $n \times n$ real symmetric matrices, $A = [a_{ij}]$, associated to $G$ where, when $i\neq j$, $a_{ij} \neq 0$ if and only if $ij$ is an edge in $G$ and the main diagonal is free to be chosen. For any square matrix, $A$, let $q(A)$ equal the number of distinct eigenvalues of $A$. The minimum number of distinct eigenvalues of...
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Alexander Guterman (Bar-Ilan University)5/19/26, 3:45 PMCombinatorial Matrix TheoryMinisymposium Talk
The first results on transformations preserving matrix invariants is due to Frobenius. This result describes the structure of linear maps $T$ preserving the determinant function, i.e., $\det X = \det T(X)$ for all $X$. Later on there were several extension of this result which are due to Diedonnie, Schur, Dynkin and others.
In 1913 Cullis and then in 1966 independently Radi\'c...
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Elizabeth Dinkelman (George Mason University)5/19/26, 4:10 PMCombinatorial Matrix TheoryMinisymposium Talk
The polytope $ASM_n$, the convex hull of the $n\times n$ alternating sign matrices, was introduced by Striker and by Behrend and Knight. A face of $ASM_n$ corresponds to an elementary flow grid defined by Striker, and each elementary flow grid determines a doubly directed graph defined by Brualdi and Dahl. We show that a face of $ASM_n$ is symmetric if and only if its doubly directed graph...
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Bryan Shader (University of Wyoming)5/19/26, 4:35 PMCombinatorial Matrix TheoryMinisymposium Talk
This talk discusses problem of determining the minumum number of nonzero entries in a pair of matrices (A,A^(-1)) for A in various families of matrices (e.g. irreducible, fully indecomposable, primitive, positive definite, orthogonal, symmetric with connected graph, irreducible covariance matrices). Some of the results are from work with H. Gupta, L. Hogben and T. Wong.
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Sudipta Mallik (Marshall University)5/19/26, 5:00 PMCombinatorial Matrix TheoryMinisymposium Talk
Given an integer k, deciding whether a graph has a clique of size k is an NP-complete problem. Wilf's inequality provides a spectral lower bound for the clique number (i.e., the order of a largest clique) in terms of the largest adjacency eigenvalue. In 2018, Elphick and Wocjan conjectured a stronger spectral bound using positive adjacency eigenvalues. We introduce a spectral bound using...
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Geir Dahl5/21/26, 2:00 PMCombinatorial Matrix TheoryMinisymposium Talk
We introduce a new rank concept for $(0,\pm 1)$-matrices, called the $\pm$-rank of a $(0,\pm 1)$-matrix. This ``generalizes'' the binary rank and the term rank of (0,1)-matrices. We establish several inequalities relating the different ranks, including ordinary real rank. Moreover, the $\pm$-rank is discussed for certain classes of $(0,\pm 1)$-matrices, such as alternating sign matrices...
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Enide Andrade (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Portugal)5/21/26, 2:25 PMCombinatorial Matrix TheoryMinisymposium Talk
Let $G$ be a mixed graph and let $(H_1, H_2)$ be an ordered pair of mixed graphs whose orders coincide with the order and size of $G$, respectively. We introduce the subdivision mixed graph $S(G)$ and the $(H_1,H_2)$-merged subdivision mixed graph. We investigate the Hermitian spectrum and the Hermitian energy of these graphs, deriving spectral properties that relate merged subdivision mixed...
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Aditya Khanna (Virginia Tech)5/21/26, 2:50 PMCombinatorial Matrix TheoryMinisymposium Talk
Let $A = (A_n)_{n\geq 0}$ and $B = (B_n)_{n\geq 0}$ be families of "recursive'' rectangular matrices, that is, the entries of $A_n$ can be expressed as a linear combination of entries of $A_m$ for $m < n$, and similarly for $B$. Such families arise in algebraic combinatorics as change-of-basis matrices between the basis of symmetric functions and their generalizations. The entries of such...
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