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A factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (S
Externí odkaz:
http://arxiv.org/abs/2308.12399
The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint
Externí odkaz:
http://arxiv.org/abs/2306.05039
Autor:
Abiad, Aida, Fallat, Shaun M., Kempton, Mark, Levene, Rupert H., Oblak, Polona, Šmigoc, Helena, Tait, Michael, Meulen, Kevin Vander
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 join of a connected graph with a path. We then focus on bordering a mat
Externí odkaz:
http://arxiv.org/abs/2303.07949
The inverse eigenvalue problem of a graph $G$ is the problem of characterizing all lists of eigenvalues of real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of $G$. The strong spectral property is a powerful tool in
Externí odkaz:
http://arxiv.org/abs/2302.01670
Autor:
Fallat, Shaun M., Hall, H. Tracy, Levene, Rupert H., Meyer, Seth A., Nasserasr, Shahla, Oblak, Polona, Šmigoc, Helena
If $G$ is a graph and $\mathbf{m}$ is an ordered multiplicity list which is realizable by at least one symmetric matrix with graph $G$, what can we say about the eigenvalues of all such realizing matrices for $\mathbf{m}$? It has sometimes been tempt
Externí odkaz:
http://arxiv.org/abs/2301.11073
Autor:
Kirkland, Stephen, Šmigoc, Helena
A celebrated result of Karpelevi\v c describes $\Theta_n,$ the collection of all eigenvalues arising from the stochastic matrices of order $n.$ The boundary of $\Theta_n$ consists of roots of certain one-parameter families of polynomials, and those p
Externí odkaz:
http://arxiv.org/abs/2110.01040
Publikováno v:
Linear algebra and its applications, vol. 665 (2023)
The Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization) is a factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are required to be
Externí odkaz:
http://arxiv.org/abs/2106.14437
We show that every $0$-$1$ multiplicity matrix for a simple graph $G$ is generically realisable for $G$. In particular, every multiplicity matrix for a path is generically realisable. We use this result to provide several families of joins of graphs
Externí odkaz:
http://arxiv.org/abs/2103.04587
Publikováno v:
In Linear Algebra and Its Applications May 2024
We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs $G$ and $H$ to be the pattern of an orthogonal symmetric matrix, or equivalently, for the mi
Externí odkaz:
http://arxiv.org/abs/2012.12694