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Carlos Hoppen

Publications and source records attributed to Carlos Hoppen.

At least 19 recordsLinked to original sources
arXiv2026

Separating path systems for cubic graphs and for complete bipartite graphs

A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of such a system is the so called strong separation number of $G$. We prove that the strong separation number of every $2$-degenerate graph on $n$ vertices is at most $n$. Using this, we also provide upper bounds for the strong separation number of subcubic graphs, planar graphs, and planar bipartite graphs. On the other hand, we prove that the strong separation number a complete bipartite graph $K_{a,b}$ is at least $b$ if $a<b/2$ and at least $(\sqrt{6(b/2)+4}-2)a$ if $b/2\leq a\leq b$, and we provide a construction that attains the former bound.

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arXiv2026

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.

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arXiv2025

Multi-Community Spectral Clustering for Geometric Graphs

In this paper, we consider the soft geometric block model (SGBM) with a fixed number $k \geq 2$ of homogeneous communities in the dense regime, and we introduce a spectral clustering algorithm for community recovery on graphs generated by this model. Given such a graph, the algorithm produces an embedding into $\mathbb{R}^{k-1}$ using the eigenvectors associated with the $k-1$ eigenvalues of the adjacency matrix of the graph that are closest to a value determined by the parameters of the model. It then applies $k$-means clustering to the embedding. We prove weak consistency and show that a simple local refinement step ensures strong consistency. A key ingredient is an application of a non-standard version of Davis-Kahan theorem to control eigenspace perturbations when eigenvalues are not simple. We also analyze the limiting spectrum of the adjacency matrix, using a combination of combinatorial and matrix techniques.

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arXiv2025

Maximum number of edge colorings avoiding rainbow copies of $K_4$

In this paper we show that for $r\geq 12$ and any sufficiently large $n$-vertex graph $G$ the number of $r$-edge-colorings of $G$ with no rainbow $K_4$ is at most $r^{ex(n,K_4)}$, where $ex(n,K_4)$ denotes the Turán number of $K_4$. Moreover, $G$ attains equality if and only if it is the Turán graph $T_3(n)$. The bound on the number of colors $r\geq 12$ is best possible. It improves upon a result of H. Lefmann, D.A. Nolibos, and the second author who showed the same result for $r \geq 5434$ and it confirms a conjecture by Gupta, Pehova, Powierski and Staden.

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arXiv2025

On the minimum number of eigenvalues of trees of diameter seven

The underlying graph $G$ of a symmetric matrix $M=(m_{ij})\in \mathbb{R}^{n\times n}$ is the graph with vertex set $\{v_1,\ldots,v_n\}$ such that a pair $\{v_i,v_j\}$ with $i\neq j$ is an edge if and only if $m_{ij}\neq 0$. Given a graph $G$, let $q(G)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $G$. A symmetric matrix $M$ is said to be a realization of $q(G)$ if it has underlying graph $G$ and $q(G)$ distinct eigenvalues. In the case of trees, a paper by Johnson and Saiago [Johnson, C.R, and Saiago, C.M, Diameter Minimal Trees, Linear and Multilinear Algebra 64(3) (2015), 557--571.] proposed an approach by which realizations of large trees are constructed from realizations of smaller trees with the same diameter, known as seeds, which has proved to be very successful. In this paper, we discuss realizations of $q(T)$ for trees of diameter seven based on the seed that defines it, correcting a result in the aforementioned paper.

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arXiv2023

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.

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arXiv2022

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 Turán graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.

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arXiv2021

Efficient diagonalization of symmetric matrices associated with graphs of small treewidth

Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in an arbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set $\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if and only if $m_{ij} \neq 0$. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to $M$. If $G$ is given with a tree decomposition $\mathcal{T}$ of width $k$, then this can be done in time $O(k|\mathcal{T}| + k^2 n)$, where $|\mathcal{T}|$ denotes the number of nodes in $\mathcal{T}$. Among other things, this allows one to compute the determinant, the rank and the inertia of a symmetric matrix in time $O(k|\mathcal{T}| + k^2 n)$.

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arXiv2021

An extension of the rainbow Erdős-Rothschild problem

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called $\mathcal{P}_{k,s}$-free $r$-colorings. We show that, for large $n$ and $r \geq r_0(k,s)$, the $(k-1)$-partite Turán graph $T_{k-1}(n)$ on $n$ vertices yields the largest number of $\mathcal{P}_{k,s}$-free $r$-colorings among all $n$-vertex graphs, and that it is the unique graph with this property.

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arXiv2021

Counting orientations of graphs with no strongly connected tournaments

Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.

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arXiv2020

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 the dampening effect of isolation on the dissemination of the disease.

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arXiv2020

The rainbow Erdős-Rothschild problem for the Fano plane

The Fano plane is the unique linear 3-uniform hypergraph on seven vertices and seven hyperedges. It was recently proved that, for all $n \geq 8$, the balanced complete bipartite 3-uniform hypergraph on $n$ vertices, denoted by $B_n$, is the 3-uniform hypergraph on $n$ vertices with the largest number of hyperedges that does not contain a copy of the Fano plane. For sufficiently large $r$ and $n$, we show that $B_n$ admits the largest number of $r$-edge colorings with no rainbow copy of the Fano plane.

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arXiv2020

A note on a stability result for the Fano plane

In this note, we adapt the Keevash-Sudakov proof of the (Turán) Stability Theorem for the Fano plane to find an explicit dependency between the parameters $\varepsilon$ and $δ$. This is useful in the solution of a multicolored version for hypergraphs of an extremal problem about edge-colorings, known as the Erdős-Rothschild problem, which may be considered for the Fano plane.

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arXiv2020

Asymptotic bounds on total domination in regular graphs

We find new upper bounds on the size of a minimum totally dominating set for random regular graphs and for regular graphs with large girth. These bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and Wormald [Local algorithms, regular graphs of large girth, and random regular graphs. Combinatorica 38(3) (2018), 619-664].

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arXiv2019

The spectrum of a class of uniform hypergraphs

A generalized power hypergraph $\mathcal{H}^k_s$ is obtained from a base hypergraph $\mathcal{H}$ by means of some simple edge-expansion operations. Kang, Liu, Qi and Yuan [8] proved that the nonzero eigenvalues of $\mathcal{H}$ give rise to nonzero eigenvalues of $\mathcal{H}^k_s$. In this paper we show that all nonzero eigenvalues of $\mathcal{H}^k_s$ may be computed from the eigenvalues of its base hypergraph $\mathcal{H}$ and of its subgraphs. To prove this, we derive spectral results about edge-expansion operations that may be interesting on their own sake.

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arXiv2017

Eigenvalue location in graphs of small clique-width

Finding a diagonal matrix congruent to $A - cI$ for constants $c$, where $A$ is the adjacency matrix of a graph $G$ allows us to quickly tell the number of eigenvalues in a given interval. If $G$ has clique-width $k$ and a corresponding $k$-expression is known, then diagonalization can be done in time $O(\text{poly}(k) n)$ where $n$ is the order of $G$.

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