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Rui Duarte

Publications and source records attributed to Rui Duarte.

18 recordsLinked to original sources
arXiv2025

A family of regular integral graphs and its application to the $n$-Queens' graph

A family of regular integral graphs introduced in [I.F.S. Costa, The $n$-Queens graph and its generalizations, Ph.D. Thesis, University of Aveiro 2024], denoted by ${\cal T}(n)$ and herein called triangular graphs, is analysed. In this analysis, the consistent structure of the graph spectra and the patterns of the corresponding eigenvectors are highlighted. The properties of these graphs are examined and applied to the decomposition of the $n$-Queens' graph into three distinct families: a family of a single graph whose components are two triangular graphs, ${\cal T}(n)$ and ${\cal T}(n-1)$, a family of a single graph whose components are cliques and a family of complete bipartite graphs. Finally, using Weyl's inequalities, we introduce some techniques to establish lower and upper bounds on the eigenvalues of the $n$-Queens' graph.

math.CO↗
arXiv2024

Generating functions of lattice paths

We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently introduced lattice paths. For every type, we consider the respective generating function. In fact, through our approach (via Riordan arrays), various entries of the On-Line Encyclopedia of Integer Sequences are unified, clarified, and simplified.

math.CO↗
arXiv2023

The number of Prime Parking Functions

A parking function of length $n$ is prime if we obtain a parking function of length $n-1$ by deleting one 1 from it. In this note we give a new direct proof that the number of prime parking functions of length $n$ is $(n-1)^{n-1}$. This proof leads to a new interpretation, in close terms to the definition of parking function.

math.CO↗
arXiv2023

Integer eigenvalues of the $n$-Queens graph

The $n$-Queens graph, $\mathcal{Q}(n)$, is the graph obtained from a $n\times n$ chessboard where each of its $n^2$ squares is a vertex and two vertices are adjacent if and only if they are in the same row, column or diagonal. In a previous work the authors have shown that, for $n\ge4$, the least eigenvalue of $\mathcal{Q}(n)$ is $-4$ and its multiplicity is $(n-3)^2$. In this paper we prove that $n-4$ is also an eigenvalue of $\mathcal{Q}(n)$ and its multiplicity is at least $\frac{n+1}{2}$ or $\frac{n-2}{2}$ when $n$ is odd or even, respectively. Furthermore, when $n$ is odd, it is proved that $-3,-2\ldots,\frac{n-11}{2}$ and $\frac{n-5}{2},\ldots,n-5$ are additional integer eigenvalues of $\mathcal{Q}(n)$ and a family of eigenvectors associated with them is presented. Finally, conjectures about the multiplicity of the aforementioned eigenvalues and about the non-existence of any other integer eigenvalue are stated.

math.CO↗
arXiv2022

Sharp bounds on the least eigenvalue of a graph determined from edge clique partitions

Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the $n$-Queens' graph $\mathcal{Q}(n)$ is equal to $-4$ for every $n \ge 4$ and it is also proven that the multiplicity of this eigenvalue is $(n-3)^2$. Additionally, some results on the edge clique partition graph parameters are obtained.

math.CO↗
arXiv2021

Deep Neural Networks for Active Wave Breaking Classification

Wave breaking is an important process for energy dissipation in the open ocean and coastal seas. It drives beach morphodynamics, controls air-sea interactions, determines when ship and offshore structure operations can occur safely, and influences on the retrieval of ocean properties from satellites. Still, wave breaking lacks a proper physical understanding mainly due to scarce observational field data. Consequently, new methods and data are required to improve our current understanding of this process. In this paper we present a novel machine learning method to detect active wave breaking, that is, waves that are actively generating visible bubble entrainment in video imagery data. The present method is based on classical machine learning and deep learning techniques and is made freely available to the community alongside this publication. The results indicate that our best performing model had a balanced classification accuracy score of ~90% when classifying active wave breaking in the test dataset. An example of a direct application of the method includes a statistical description of geometrical and kinematic properties of breaking waves. We expect that the present method and the associated dataset will be crucial for future research related to wave breaking in several areas of research, which include but are not limited to: improving operational forecast models, developing risk assessment and coastal management tools, and refining the retrieval of remotely sensed ocean properties.

physics.ao-ph↗
arXiv2020

Spectral properties of the $n$-Queens' Graphs

The $n$-Queens' graph, $\mathcal{Q}(n)$, is the graph associated to the $n \times n$ chessboard (a generalization of the classical $8 \times 8$ chessboard), with $n^2$ vertices, each one corresponding to a square of the chessboard. Two vertices of $\mathcal{Q}(n)$ are adjacent if and only if they are in the same row, in the same column or in the same diagonal of the chessboard. After a short overview on the main combinatorial properties of $\mathcal{Q}(n)$, its spectral properties are investigated. First, a lower bound on the least eigenvalue of an arbitrary graph is obtained using clique edge partitions and a sufficient condition for this lower bound be attained is deduced. For the particular case of $\mathcal{Q}(n)$, we prove that for every $n$, its least eigenvalue is not less than $-4$ and it is equal to $-4$ with multiplicity $(n-3)^2$, for every $n \ge 4$. Furthermore, $n-4$ is also an eigenvalue of $\mathcal{Q}(n)$, with multiplicity at least $\frac{n-2}{2}$ when $n$ is even and at least $\frac{n+1}{2}$ when $n$ is odd. A conjecture about the integer eigenvalues of $\mathcal{Q}(n)$ is presented. We finish this article with an algorithm to determine an equitable partition of the $n$-Queens' graph, $\mathcal{Q}(n)$, for $n \ge 3$, concluding that such equitable partition has $\frac{(\lceil n/2\rceil+1)\lceil n/2\rceil}{2}$ cells.

math.CO↗
arXiv2019

Pak-Stanley labeling of the $m$-Catalan hyperplane arrangement

We characterize in simple terms the Pak-Stanley labels $\lambda(R)$ of the regions $R$ of the $m$-Catalan arrangement. We also propose a simple algorithm that returns $R$ from $\lambda(R)$. Finally, we characterize in close terms the labels of the relatively bounded regions.

math.CO↗
arXiv2018

Partial Parking Functions

We characterise the Pak-Stanley labels of the regions of a family of hyperplane arrangements that interpolate between the Shi arrangement and the Ish arrangement.

math.CO↗
arXiv2017

Making data center computations fast, but not so furious

We propose an aggressive computational sprinting variant for data center environments. While most of previous work on computational sprinting focuses on maximizing the sprinting process while ensuring non-faulty conditions, we take advantage of the existing replication in data centers to push the system beyond its safety limits. In this paper we outline this vision, we survey existing techniques for achieving it, and we present some design ideas for future work in this area.

cs.DC↗
arXiv2017

Between Ish and Shi

We introduce a new family of hyperplane arrangements in dimension $n\geq3$ that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions - the connected components of the complement of the union of the hyperplanes - which can be bijectively labeled with the Pak-Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.

math.CO↗
arXiv2016

The number of parking functions with center of a given length

Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$ vertices are visited before backtracking. Let $R$ be the set of trees with this property. We count the number of elements of $R$. For this purpose, we first consider a bijection, due to Parkinson, Yang and Yu, that maps $R$ onto the set of parking function with center (defined by the authors in a previous article) of size $r$. A second bijection maps this set onto the set of parking functions with run $r$, a property that we introduce here. We then prove that the number of length $n$ parking functions with a given run is the number of length $n$ rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length $n$ rook words with run $r$, which is the answer to our initial question.

math.CO↗
arXiv2015

Note on the bijectivity of the Pak-Stanley labelling

This article has the sole purpose of presenting a simple, self-contained and direct proof of the fact that the Pak-Stanley labeling is a bijection. The construction behind the proof is subsumed in a forthcoming paper [R. Duarte and A. Guedes de Oliveira, The braid and the Shi arrangements and the Pak-Stanley labeling, Eur. J. Combinatorics, in press.], but an actual self-contained proof is not explicitly included in that paper.

math.CO↗
arXiv2013

Short note on the convolution of binomial coefficients

We know [Rui Duarte and Ant\'onio Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n,i,j$ and for every real number $\ell$, $$ \sum_{i+j=n} \binom{2i-\ell}{i} \binom{2j+\ell}{j} = \sum_{i+j=n}\binom{2i}{i} \binom{2j}{j}, $$ which is well-known to be $4^n$. We extend this result by proving that, indeed, $$ \sum_{i+j=n} \binom{ai+k-\ell}{i} \binom{aj+\ell}{j} = \sum_{i+j=n} \binom{ai+k}{i} \binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value.

math.CO↗
arXiv2012

New developments of an old identity

We give a direct combinatorial proof of a famous identity, $$ \sum_{i+j=n} m{2i}{i} \binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the paper by presenting in explicit form the ordinary generating function of the sequence $(\strut\binom{2n+k}{n})_{n\in\mathds{N}_0}$, where $k\in\mathds{R}$.

math.CO↗
arXiv2011

Constructions of bipartite and bipartite-regular hypermaps

A hypermap is bipartite if its set of flags can be divided into two parts A and B so that both A and B are the union of vertices, and consecutive vertices around an edge or a face are contained in alternate parts. A bipartite hypermap is bipartite-regular if its set of automorphisms is transitive on A and on B. In this paper we see some properties of the constructions of bipartite hypermaps described algebraically by Breda and Duarte which generalize the construction induced by the Walsh representation of hypermaps. As an application we show that all surfaces have bipartite-regular hypermaps.

math.CO↗
arXiv2006

Bipartite-uniform hypermaps on the sphere

A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that ``neighbouring'' hypervertices have different colours. It is bipartite-uniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, elements have common valencies. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hypermap is regular. In this paper we classify the bipartite-uniform hypermaps on the sphere (up to duality). Two constructions of bipartite-uniform hypermaps are given. All bipartite-uniform spherical hypermaps are shown to be constructed in this way. As a by-product we show that every bipartite-uniform hypermap on the sphere is bipartite-regular. We also compute their irregularity group and index, and also their closure cover and covering core.

math.CO↗