Nair Abreu

Nair Abreu
Federal University of Rio de Janeiro | UFRJ · Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa de Engenharia (COPPE)

Phd

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87
Publications
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Publications

Publications (87)
Preprint
In this paper, we establish the relation between classic invariants of graphs and their integer Laplacian eigenvalues, focusing on a subclass of chordal graphs, the strictly chordal graphs, and pointing out how their computation can be efficiently implemented. Firstly we review results concerning general graphs showing that the number of universal...
Article
Full-text available
The eigenvalues of a graph are those of its adjacency matrix. Recently, Cioabă, Haemers and Vermette characterized all graphs with all but two eigenvalues equal to −2 and 0. In this article, we extend their result by characterizing explicitly all graphs with all but two eigenvalues in the interval [−2, 0]. Also, we determine among them those that a...
Article
In this paper, structural properties of chordal graphs are studied, establishing a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a...
Article
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is non orthogonal to the associated eigenspace. This paper explores some new aspects of the study of main eigenvalues of graphs, investigating specifically cones over strongly regular graphs and graphs for which the least eigenvalue is non-main. In this case, we...
Article
We present a characterization of graphs which are simultaneously block and indifference graphs. Some structural and spectral properties of the class are depicted and their interconnection is shown. We show an O(n) representation which allows us to count the number of elements of the class. Regarding spectral properties, we prove that a large subcla...
Preprint
In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we s...
Article
Let G be a simple graph. A set of pairwise disjoint edges is a matching of G. The matching polytope of G, denoted M(G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G, SMP(G) or, even simpler,...
Article
The matching polytope of a graph G, denoted by M(G), is the convex hull of the set of the incidence vectors of the matchings G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of G(M(G)) is equal to...
Article
The matching polytope of a graph G, denoted by ℳ (G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph (ℳ (G)), whose vertices and edges are the vertices and edges of ℳ (G), is the skeleton of the matching polytope of G. In this paper, for an arbitrary graph, we prove that the minimum degree of (ℳ (G)) is equa...
Article
In this paper, we introduce a new invariant that supports an accurate evaluation of the connectivity of graphs belonging to some subclasses of chordal graphs, allowing the establishment of a total ordering of the elements of the class. It is based on the minimal vertex separators of the graph, and, as so, its computation is performed in linear time...
Article
We determine a closed-form expression for the fifth characteristic coefficient of the power of a path. To arrive at this result, we establish the number of 4-cycles in the graph by means of their structural properties. The method developed might be applied to other well-structured graph classes in order to count 4-cycles or modified to count cycles...
Article
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In honour of Dragoš Cvetković on the occasion of his 75th birthday. Let G be a graph on n vertices and G its complement. In this paper, we prove a Nordhaus-Gaddum type inequality to the second largest eigenvalue of a graph G, λ2(G when G is a graph with girth at least 5. Also, we show that the bound above is tight. Besides, we prove that this resul...
Article
Given a tree T, we consider a pair of vertices (u, v) where u is a centroid of T, v is a characteristic vertex of T, and such that the distance between them, denoted d(u, v), is smallest over all such pairs. We define and where the maximum is taken over all trees T on n vertices. Analogous definitions are also given for and . We show that for each...
Article
The convex hull of the set of the incidence vectors of the matchings of a graph G is the matching polytope of the graph, M(G). The graph whose vertices and edges are the vertices and edges of M(G) is the skeleton of the matching polytope of G, denoted G(M(G)). Since the number of vertices of G(M(G)) is huge, the structural properties of these graph...
Article
Consider two graphs $G$ and $H$. Let $H^k[G]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k[G]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k[G]$ and $H^k$ for $G$ and $H$ arbitrary. Particular...
Article
Let G be a graph with n vertices and e(G) edges. The signless Laplacian of G, denoted by Q(G), is given by Q(G)=D(G)+A(G), where D(G) and A(G) are the diagonal matrix of its vertex degree and A(G) is the adjacency matrix. Let q1(G),…,qn(G) be the eigenvalues of Q(G) in non-increasing order and let Tk(G)=∑i=1kqi(G) be the sum of the k largest signle...
Article
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An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of h...
Technical Report
Full-text available
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic...
Article
Full-text available
In this paper, trees with fixed diameter and any number of vertices are investigated. A subclass of trees with diameter $2k$ is introduced, the \textit{diameter path trees} ($dp$-trees). Two subclasses of $dp$-trees are defined in which we characterize the elements that maximize the algebraic connectivity. Also, it is proved that if any tree maximi...
Article
Full-text available
A bug Bug p,r1r2 is a graph obtained from a complete graph Kp by deleting an edge uv and attaching the paths Pri and Pr2 by one of their end vertices at u and v, respectively. Let Q(G) be the signless Laplacian matrix of a graph G and q1(G) be the spectral radius of Q(G). It is known that the bug maximizes q1(G) among all graphs G of order n and di...
Article
Full-text available
A complete caterpillar is a caterpillar in which each internal vertex is a quasi-pendent vertex. In this paper, in the class of all complete caterpillars on n vertices and diameter d, the caterpillar attaining the largest Laplacian index is determined. In addition, it is proved that this caterpillar also attains the largest adjacency index.
Article
For a fixed tree T with n vertices the corresponding acyclic Birkhoff polytope Ωn(T)Ωn(T) consists of doubly stochastic matrices having support in positions specified by T . This is a face of the Birkhoff polytope ΩnΩn (which consists of all n×nn×n doubly stochastic matrices). The skeleton of Ωn(T)Ωn(T) is the graph where vertices and edges corresp...
Article
Full-text available
A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which non- regular bipartite integral graphs with maximum degree four...
Article
Every acyclic Birkhoff polytope is represented by a bicolored tree. In this paper we use the concept of $T$-component of a tree in order to cover it. In addition, the definitions of $T$- edge cover (respectively, $T$-vertex cover) subgraphs and of complementary coverage by vertices( edges) are introduced. Some consequences related to the dimension...
Article
Full-text available
A graph is regular if every vertex is of the same degree. Otherwise, it is an irregular graph. Although there is a vast literature devoted to regular graphs, only a few papers approach the irregular ones. We have found four distinct graph invariants used to measure the irregularity of a graph. All of them are determined through either the average o...
Article
Full-text available
This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number r, then the largest eigenvalue q(G) of the matrix Q=A+D satisfies...
Conference Paper
Full-text available
In this article, we construct graphs that are simultaneously integral, Laplacian integral and signless Laplacian integral. The graphs with this property, known in the literature, are regular or bipartite, unless by a few exceptions. We obtain infinite families of such graphs, that are neither regular nor bipartite, from join of regular graphs
Article
Full-text available
The second smallest Laplacian eigenvalue of a graph G is called algebraic connectivity, denoted a (G). The ordering of trees via this graph invariant is frequently studied in the literature. In this paper, we present a new invariant, the Internal Degree Sequence (IDS), that also supports an accurate evaluation of the connectivity of trees. We compa...
Article
Full-text available
Consider the Laplacian and signless Laplacian spectrum of a graph G of order n, with k pairwise co-neighbor vertices. We prove that the number of shared neighbors is a Laplacian and a signless Laplacian eigenvalue of G with multiplicity at least k − 1. Additionally, considering a connected graph GkGk with a vertex set defined by the k pairwise co-n...
Article
Let q(G) denote the spectral radius of the signless Laplacian matrix of a graph G, also known as the Q-index of G. The aim of this note is to study a general extremal problem: How large can q(G) be when G belongs to an abstract graph property? Even knowing very little about the graph property, this paper shows that useful conclusions about the asym...
Article
The spread of a real and symmetric matrix M, denoted s M , is given by the difference between the largest eigenvalue of M (also known as M-index) and its smallest eigenvalue. Our paper relates s M , where M=A(G),L(G),Q(G) is the adjacency, Laplacian or signless Laplacian matrices of a graph G to the chromatic number of G, χ(G). The main result of t...
Article
The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degr...
Article
Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless...
Article
Full-text available
The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many applications in several areas such as operational research, parallel and distributed computing, and combinatorial data analysis. Other optimization combinatorial problems such as the traveling salesman problem, maximal clique, isomorphism an...
Conference Paper
This work presents a computer program developed for the definition of Deck Accommodation Arrangements of ships and craft that allocates spaces in an automated process, in order to help the designer with options and initial ideas for its definition or to optimize arrangements propositions. This optimization process is usually done based on experienc...
Article
The spread s(M) of an n×n complex matrix M is s(M)=maxij|λi-λj|, where the maximum is taken over all pairs of eigenvalues of M, λi,1⩽i⩽n [E. Jiang, X. Zhan, Lower bounds for the spread of a hermitian matrix, Linear Algebra Appl. 256 (1997) 153–163; J. Kaarlo Merikoski, R. Kumar, Characterizations and lower bounds for the spread of a normal matrix,...
Article
The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singul...
Article
In this article, we characterize all signless Laplacian integral graphs (here called QQ-integral graphs) in the following classes: complete split graphs, multiple complete split-like graphs, extended complete split-like graphs, multiple extended split-like graphs. All these graphs were defined by Hansen et al. [P. Hansen, H. Melot, D, Stevanović, I...
Article
Full-text available
A caterpillar is a tree in which the removal of all pendent vertices make it a path. We consider two classes of caterpillars. We present an ordering of caterpillars by algebraic connectivity in one of them and find one that maximizes the algebraic connectivity in the other class.
Article
Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees Δ=d1≥d2≥⋯≥dn=δ. The average degree of the neighbor of vertex vi is . Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)−A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G...
Article
The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24 (2003) pp. 413 − 430, about the possibility of finding split non-threshold graphs that are Laplacian integral, i.e. graphs for which the eigenvalues of the corresponding Laplacian matrix are integers. Using Kronecker products, balanced incomplet...
Article
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d⩾3 and n>2(d-1) be given. Let p=[p1,p2,…,pd-1] with p1⩾1,p2⩾1,…,pd-1⩾1. Let C(p) be the caterpillar obtained from the stars Sp1,Sp2,…,Spd-1 and the path Pd-1 by identifying the root of Spi with the i-vertex of Pd-1. LetC=C(p):p1+p2+⋯+pd-1=n-d+1.We prove that...
Article
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. The Laplacian (respectively, the signless Laplacian) energy of $G$ is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the verte...
Article
Full-text available
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d ≥ 3 and n ≥ 6 be given. Let Pd−1 be the path of d − 1 vertices and Sp be the star of p + 1 vertices. Let p = [p1, p2, ..., pd−1] such that p1 ≥ 1, p2 ≥ 1, ..., pd−1 ≥ 1. Let C (p) be the caterpillar obtained from the stars Sp1 , Sp2 , ...,Spd−1 and the path...
Article
Let G be a connected graph with two nonadjacent vertices and G′ be the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′, respectively. Hence, the sum of the eigenvalues of Q′ is the sum of the eigenvalues of Q plus 2. Si...
Article
A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d⩾3 and n⩾6 be given. Let Pd−1 be the path on d−1 vertices and K1,p be the star of p+1 vertices. Let p=[p1,p2,…,pd−1] such that ∀i,1⩽i⩽d−1,pi. Let C(p) be the caterpillar obtained from d−1 stars K1,pi and the path Pd−1 by identifying the root of K1,pi with the...
Article
The energy of a simple graph G is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equ...
Article
Full-text available
For a graph G with n vertices and m edges, with ordinary spectrum λi, i = 1, 2, . . ., n, and with Laplacian spectrum μi, i = 1, 2, . . ., n, the energy and the Laplacian energy are defined as E(G) = ∑i=1n |λi| and LE(G) = ∑i=1n |μi - 2m/n|, respectively. It is known that E(G) = LE(G) if G is regular. We now show that there are non-regular graphs w...
Article
The energy of a simple graph G is equal to the sum of the absolute values of the eigenvalues of its adjacency matrix. The Laplacian energy of G, recently introduced in the literature, is an analogous graph invariant defined as a function of the eigenvalues of the Laplacian matrix and the average degree of G. We investigate the Laplacian energy of t...
Article
This paper is a survey of the second smallest eigenvalue of the Laplacian of a graph G, best-known as the algebraic connectivity of G, denoted a(G). Emphasis is given on classifications of bounds to algebraic connectivity as a function of other graph invariants, as well as the applications of Fiedler vectors (eigenvectors related to a(G)) on trees,...
Article
Let Gn,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of Gn,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and...
Article
We establish a useful correspondence between the closed walks in regular graphs and the walks in infinite regular trees, which, after counting the walks of a given length between vertices at a given distance in an infinite regular tree, provides a lower bound on the number of closed walks in regular graphs. This lower bound is then applied to reduc...
Article
The idea of equilibrium of a graph G, initially applied to maximal outerplanar graphs (mops), was conceived to observe how the vertex degree distribution affects the average degree of the graph, d(G). In this work, we formally extend the concept to graphs in general. From d(G), two new parameters are introduced - the top and the gap of G, sustainin...
Article
Full-text available
The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph parti...
Chapter
The algebraic connectivity a(G) of a graph G = (V, E) is the second smallest eigenvalue of its Laplacian matrix. Using the AutoGraphiX (AGX) system, extremal graphs for algebraic connectivity of G in function of its order n = |V| and size m = |E| are studied. Several conjectures on the structure of those graphs, and implied bounds on the algebraic...
Article
For some a and b positive rational numbers, a simple graph with n vertices and m=an−b edges is an (a,b)-linear graph, when n>2b. We characterize non-empty classes of (a,b)-linear graphs and determine those which contain connected graphs. For non-empty classes, we build sequences of (a,b)-linear graphs and sequences of connected (a,b)-linear graphs....
Article
Full-text available
O Problema Quadrático de Alocação, PQA, um dos mais difíceis da classe NP-hard, modela diversas aplicações em diferentes áreas como pesquisa operacional, computação paralela e análise estatística de dados discretos. Além disso, problemas conhecidos como o do caixeiro viajante, o da clique maximal, o de particionamento e o de isomorfismo de grafos p...
Article
We characterize the (a,b)-linear class of graphs as a function of the average degree of G. Besides, we prove, under certain conditions, that every graph with n vertices and m edges belongs to only one (a,b)-linear class. When G is a forest, we also prove that a=1 and b is the number of its trees. If n and m are given, Hakimi’s algorithm generates g...
Article
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O Problema Quadrático de Alocação, PQA, pode ser abordado através de uma relaxação na forma do Problema de Alocação Linear, PAL. Introduzimos um poset (conjunto parcialmente ordenado) no conjunto das soluções lineares que nos permite comparar também os custos das soluções do problema quadrático, sem o conhecimento prévio das matrizes que definem se...
Article
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Este texto é um pequeno e merecido tributo ao Professor Nelson Maculan Filho por ocasião do seu 60masculine aniversário, em reconhecimento às suas grandes contribuições à Universidade brasileira. Como amigos, colaboradores e ex-alunos, ficamos sensibilizados e honrados com o convite feito pelo Professor Jayme Luiz Szwarcfiter para escrever este art...
Article
In this work, we introduce the variance expression for quadratic assignment problem (QAP) costs. We also define classes of QAP instances, described by a common linear relaxation form. The use of the variance in these classes leads to the study of isomorphism and allows for a definition of a new difficulty index for QAP instances. This index is then...
Article
In this work we deal with the characteristic polynomial of the Laplacian of a graph. We present some general results about the coefficients of this polynomial. We present families of graphs, for which the number of edges m is given by a linear function of the number of vertices n. In some of these graphs we can find certain coefficients of the abov...
Article
It is well known that the coefficients of the characteristic polynomial of a graph G are determined by the elementary subgraphs of G; in particular there are simple expressions for the four higest coefficients. Here we use an enumeration of elementary subgraphs called k-matchings to obtain expressions for the next two coefficients.
Article
We introduce families of graphs, whose number of edges is given by a linear function of the cardinality of vertices deriving from pairs of positive rational numbers. For graphs of these families, we prove certain properties related to the number of vertices whose degrees are limited to a given number. These properties generalize well-known results...
Article
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O Problema Quadrático de Alocação (PQA) pertence à classe dos problemas NP-Hard e desafia os pesquisadores tanto em sua teoria quanto em sua parte computacional. Pela sua alta complexidade muitos métodos heurísticos têm sido desenvolvidos para tentar resolvê-lo aproximadamente. A metaheurística GRASP (greedy randomized adaptive search procedures) s...
Article
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O Problema Quadrático de Alocação (PQA) pertence à classe dos problemas NP-hard. Apesar de muitos métodos heurísticos terem sido desenvolvidos visando a resolução de suas instâncias, ainda não é possível encontrar soluções ótimas para instâncias de ordem acima de 25- Uma versão do GRASP, desenvolvida por Li, Pardalos e Resende [LPR94], se mostrou b...
Article
An algebraic and combinatorial approach to the study of the Quadratic Assignment Problem produced theoretical results that can be applied to (meta) heuristics to give them information about the problem structure, allowing the construction of algorithms. In this paper those results were applied to inform a Simulated Annealing-type heuristic (which w...
Article
This paper develops an algebraic study of the travelling salesman problem (TSP) as an instance of the quadratic assignment problem (QAP). The QAP solutions can become simpler and more elegant through the use of the flux matrix, binary for TSP instances. The definition of the quotient graph with vertices in correspondence to the solutions to the pro...