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Leslie Hogben (American Institute of Mathematics, Iowa State University, Purdue University)5/18/26, 3:45 PMSpectral Graph TheoryMinisymposium Talk
The graph ${\mathcal G}(A)$ of a real symmetric matrix $n\times n$ matrix $A=[a_{ij}]$ has vertices $V=\{1,\dots,n\}$ and edges $E=\{ \{i,j\}: a_{ij}\ne 0\mbox{ and } i\ne j\}$. The set of matrices described by a graph $G$ is ${\mathcal S}(G)=\{A\in{\mathbb R}^{n\times n}:{\mathcal G}(A)=G \mbox{ and } A^\top=A\}$ and the
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Inverse Eigenvalue Problem of $G$ (IEP-$G$) is to determine all... -
Brendan Rooney (Rochester Institute of Technology)5/18/26, 4:10 PMSpectral Graph TheoryMinisymposium Talk
Given a graph $G$ on $n$ vertices, $\mathcal{S}(G)$ is the set of symmetric $n\times n$ matrices with the same off-diagonal zero pattern as the adjacency matrix $A(G)$. We say that a graph $G$ has $q(G)=2$ if there is a matrix $M\in \mathcal{S}(G)$ with exactly $2$ distinct eigenvalues. This is equivalent to the existence of an orthogonal matrix $M\in\mathcal{S}(G)$. We are interested in...
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LUIZ EMILIO ALLEM (Universidade Federal do Rio Grande do Sul)5/18/26, 4:35 PMSpectral Graph TheoryMinisymposium Talk
We study the minimum number of distinct eigenvalues $q(G)$ of threshold graphs. It is known that every threshold graph satisfies $q(G)\leq 4$, which suggests that a complete combinatorial characterization according to the values $q(G)=2,3,4$ is possible.
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In this talk, we present ongoing results toward such a characterization. -
Polona Oblak (University of Ljubljana)5/18/26, 5:00 PMSpectral Graph TheoryMinisymposium Talk
For a tree $T$ we consider the set ${\mathcal S}(T)$ of real symmetric matrices whose off-diagonal zero-nonzero pattern is equal to the pattern of the adjacency matrix of $T$. It is well known that the maximum multiplicity of an eigenvalue over matrices in ${\mathcal S}(T)$ is equal to the path cover number $P(T)$ of the tree $T$.
We present a novel decomposition of the tree into a set of...
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John Byrne (University of Delaware)5/19/26, 3:45 PMSpectral Graph TheoryMinisymposium Talk
Let $F$ be a digraph. What is the largest possible minimum outdegree on an $n$-vertex digraph which does not contain a copy of $F$? I will discuss algebraic approaches to this question and present some of my related work (joint with Michael Tait).
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Igor Balla (Leipzig University)5/19/26, 4:10 PMSpectral Graph TheoryMinisymposium Talk
In this talk, we will present a proof of the fact that Boolean matrices with bounded $\gamma_2$-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We will also discuss further structural results about Boolean matrices of bounded $\gamma_2$-norm and applications in communication...
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Shengtong Zhang (Stanford University)5/19/26, 4:35 PMSpectral Graph TheoryMinisymposium Talk
Cosine polynomials of the form $$f(x) = \cos(a_1 x) + \cos(a_2 x) + … + \cos(a_n x)$$ appear extensively in number theory and combinatorics. An old problem of Ankeny and Chowla asks: if $a_1, \cdots, a_n$ are distinct positive integers, how small must the minimum value of $f(x)$ on $[0, 2\pi]$ be? Concurrently with Benjamin Bedert, we show that the minimum value of $f(x)$ must be polynomial in...
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Emanuel Juliano (Federal University of Minas Gerais)5/19/26, 5:00 PMSpectral Graph TheoryMinisymposium Talk
In the 1980s, using the software Graffiti, Fajtlowicz made a conjecture relating the independence number of a graph with its energy, a spectral parameter introduced by Gutman (1978). By formulating the graph energy as a semidefinite program (SDP), we take a step towards Fajtlowicz's conjecture, relating the graph energy to the fractional clique covering number. As a byproduct of the SDP...
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Mr Matt Burnham (Iowa State University)5/21/26, 2:00 PMSpectral Graph TheoryMinisymposium Talk
Given a real number $q$, the $q$-Laplacian of a graph $G$ is the matrix $A+qD$ where $A$ is the adjacency matrix and $D$ the diagonal degree matrix. If the edge set of $G$ can be partitioned into edge-disjoint copies of $K_t$, then $G$ is called $K_{t}$-decomposable.
In this talk, we generalize some results from a survey paper of Cvetković, Rowlinson, and Simić about the signless...
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Isabel Byrne (University of Delaware)5/21/26, 2:25 PMSpectral Graph TheoryMinisymposium Talk
In 1985, Brouwer and Mesner proved that the vertex-connectivity of a strongly regular graph equals its valency and the only disconnecting sets of this size are point neighborhoods. In 2009, Brouwer and Koolen generalized this result to distance-regular graphs. In 1996, Brouwer conjectured that the minimum size of a disconnecting set of vertices whose removal disconnects a connected strongly...
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Vishal Gupta (University of Rochester)5/21/26, 2:50 PMSpectral Graph TheoryMinisymposium Talk
The second largest eigenvalue of a graph's adjacency matrix captures important structure and spectral properties including connectivity and expansion. From the Alon-Boppana theorem, we know that if $\theta<2\sqrt{k-1}$, then there are only finitely many $k$-regular graphs with the second largest eigenvalue at most $\theta$. This motivates the following natural question posed by Richey, Stover,...
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William Martin (Worcester Polytechnic Institute)5/21/26, 3:15 PMSpectral Graph TheoryMinisymposium Talk
Delsarte theory has been extended to the study of subsets in an increasing variety of association schemes in recent years, with many different motivations and applications. Tools originally developed for the study of error-correcting codes in the Hamming scheme and combinatorial $t$-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. The goal...
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