Carlos Hoppen
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
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Carlos Hoppen has a total of 81 co-authors. Here are the top co-authors by number of shared publications:
- Vilmar Trevisan
Instituto de Matemática e Estatística Universidade Federal do Rio Grande do Sul Porto Alegre Brasil
25 shared publications - Yoshiharu Kohayakawa
Instituto de Matemática e Estatística Universidade de Sao Paulo Sao Paulo Brazil
20 shared publications Hanno Lefmann
18 shared publications- David P. Jacobs
Clemson University
12 shared publications - Hanno Lefmann
TU Chemnitz
10 shared publications - Rudini Sampaio
Universidade Federal do Ceará
10 shared publications
Latest Publications
- Open Access
On the minimum number of eigenvalues of matrices associated with cographs
A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in E$. We prove that, for every cograph $G$, there is a matrix $M$ associated with $G$ for which the number of distinct eigenvalues is at most 4.Open AccessOn the minimum number of eigenvalues of matrices associated with cographs
A symmetric matrix $M=(m_{ij}) \in \mathbb{R}^{n \times n}$ is said to be associated with an $n$-vertex graph $G=(V,E)$ with vertex set $\{v_1,\ldots,v_n\}$ if, for every $i \neq j$, we have $m_{ij} \neq 0$ if and only if $\{v_i,v_j\}\in E$. We prove that, for every cograph $G$, there is a matrix $M$ associated with $G$ for which the number of distinct eigenvalues is at most 4. - 28 June 2025
- Published by Springer Nature in Computational and Applied Mathematics
- Vol. 44 (7) , 1-24
A self-adjusting spectral clustering algorithm
- 28 June 2025
- Published by Springer Nature in Computational and Applied Mathematics
- Vol. 44 (7) , 1-24
A self-adjusting spectral clustering algorithm
- Open AccessThe minimum number of distinct eigenvalues of a threshold graph is at most $4$In this note we show that the minimum number of distinct eigenvalues of a threshold graph is at most $4$. Moreover, given any threshold graph $G$ and any nonzero real number $λ$, we explicitly construct a matrix $M$ associated with $G$ such that DSpec$(M)\subseteq\{-λ,0,λ,2λ\}$.Open AccessThe minimum number of distinct eigenvalues of a threshold graph is at most $4$In this note we show that the minimum number of distinct eigenvalues of a threshold graph is at most $4$. Moreover, given any threshold graph $G$ and any nonzero real number $λ$, we explicitly construct a matrix $M$ associated with $G$ such that DSpec$(M)\subseteq\{-λ,0,λ,2λ\}$.
- 24 January 2024
- Published by Elsevier in Discrete Mathematics
- Vol. 347 (4)
Edge-colorings avoiding patterns in a triangle
- 24 January 2024
- Published by Elsevier in Discrete Mathematics
- Vol. 347 (4)
Edge-colorings avoiding patterns in a triangle
- 18 January 2024
- Published by University of Wyoming Libraries in The Electronic Journal of Linear Algebra
- Vol. 40, 81-139
Locating Eigenvalues of Symmetric Matrices - A Survey
We survey algorithms for locating eigenvalues of symmetric matrices taking advantage of the underlying graph. We present applications in spectral graph theory.
- 18 January 2024
- Published by University of Wyoming Libraries in The Electronic Journal of Linear Algebra
- Vol. 40, 81-139
Locating Eigenvalues of Symmetric Matrices - A Survey
We survey algorithms for locating eigenvalues of symmetric matrices taking advantage of the underlying graph. We present applications in spectral graph theory.
- Open Access
Diminimal families of arbitrary diameter
Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues.Open AccessDiminimal families of arbitrary diameter
Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues. - Open Access
- 8 December 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
- Vol. 9 (1)
Hierarchical Similarity Measure for Spectral Clustering
Hierarchical Similarity Measure for Spectral Clustering
- 8 December 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
- Vol. 9 (1)
Open AccessHierarchical Similarity Measure for Spectral Clustering
Hierarchical Similarity Measure for Spectral Clustering
- Open Access
- 8 December 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
- Vol. 9 (1)
Um problema de coloração de arestas em grafos evitando um padrão específico em triângulos
Um problema de coloração de arestas em grafos evitando um padrão específico em triângulos
- 8 December 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics
- Vol. 9 (1)
Open AccessUm problema de coloração de arestas em grafos evitando um padrão específico em triângulos
Um problema de coloração de arestas em grafos evitando um padrão específico em triângulos
- Open Access
- 8 November 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Trends in Computational and Applied Mathematics
- Vol. 23 (4) , 705-729
A Spectral Clustering Approach for the Evolution of the COVID-19 Pandemic in the State of Rio Grande do Sul, Brazil
The aim of this paper is to analyse the evolution of the COVID-19 pandemic in Rio Grande do Sul by applying graph-theoretical tools, particularly spectral clustering techniques, on weighted graphs defined on the set of 167 municipalities in the state with population 10,000 or more, which are based on data provided by government agencies and other sources. To respond to this outbreak, the state has adopted a system by which pre-determined regions are assigned flags on a weekly basis, and different measures go into effect according to the flag assigned. Our results suggest that considering a flexible approach to the regions themselves might be a useful additional tool to give more leeway to cities with lower incidence rates, while keeping the focus on public safety. Moreover, simulations show that the combination of pendulum migration and isolation data used in this paper leads to a coherent qualitative description of the evolution of the pandemic in Rio Grande do Sul. These simulations also confirm the dampening effect of isolation on the dissemination of the disease.
- 8 November 2022
- Published by Brazilian Society for Computational and Applied Mathematics (SBMAC) in Trends in Computational and Applied Mathematics
- Vol. 23 (4) , 705-729
Open AccessA Spectral Clustering Approach for the Evolution of the COVID-19 Pandemic in the State of Rio Grande do Sul, Brazil
The aim of this paper is to analyse the evolution of the COVID-19 pandemic in Rio Grande do Sul by applying graph-theoretical tools, particularly spectral clustering techniques, on weighted graphs defined on the set of 167 municipalities in the state with population 10,000 or more, which are based on data provided by government agencies and other sources. To respond to this outbreak, the state has adopted a system by which pre-determined regions are assigned flags on a weekly basis, and different measures go into effect according to the flag assigned. Our results suggest that considering a flexible approach to the regions themselves might be a useful additional tool to give more leeway to cities with lower incidence rates, while keeping the focus on public safety. Moreover, simulations show that the combination of pendulum migration and isolation data used in this paper leads to a coherent qualitative description of the evolution of the pandemic in Rio Grande do Sul. These simulations also confirm the dampening effect of isolation on the dissemination of the disease.
- Open Access
Edge-colorings avoiding patterns in a triangle
For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large, the maximum is attained by the bipartite Tur\'{a}n graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.Open AccessEdge-colorings avoiding patterns in a triangle
For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large, the maximum is attained by the bipartite Tur\'{a}n graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.