Carla Silva Oliveira
@ibge.gov.br
Escola Nacional de Ciências Estatísticas
Scopus Publications
Scopus Publications
Aida Abiad, Leonardo de Lima, Sina Kalantarzadeh, Mona Mohammadi, and Carla Oliveira
Elsevier BV
João Domingos Gomes da Silva, Carla Silva Oliveira, and Liliana Manuela G.C. da Costa
EDP Sciences
Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D(G). In 2017, Nikiforov (V. Nikiforov, Appl. Anal. Discret. Math. 11 (2017) 81–107.) defined the matrix Aα(G), as a convex combination of A(G) and D(G), the following way, Aα(G) = αA(G) + (1 − α)D(G) where α ∈ [0,1]. In this paper we present some new upper and lower bounds for the largest, second largest and the smallest eigenvalue of Aα-matrix. Moreover, extremal graphs attaining some of these bounds are characterized.
João Domingos Gomes da Silva Junior, Carla Silva Oliveira, and Liliana Manuela Gaspar Cerveira da Costa
Springer Science and Business Media LLC
Rebecca Salles, Esther Pacitti, Eduardo Bezerra, Celso Marques, Carla Pacheco, Carla Oliveira, Fabio Porto, and Eduardo Ogasawara
Springer Berlin Heidelberg
Diego Pacheco, , Carla Oliveira, Anderson Novanta, , and
University Library in Kragujevac
A.E. Brondani, F.A.M. França, and C.S. Oliveira
Elsevier BV
Abstract Let G be a connected graph of order n , A ( G ) the adjacency matrix of G and D ( G ) the diagonal matrix of the row-sums of A ( G ) . For every real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we obtain a factorization of the A α -characteristic polynomial of the some families of graphs and determine in these families the conditions about α so that A α ( G ) is positive semidefinite.
André Ebling Brondani, Francisca Andrea Macedo França, and Carla Silva Oliveira
Universidad Catolica del Norte - Chile
Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α ∈ [0, 1], Nikiforov [21] and Wang et al. [26] defined the matrices Aα(G) and Lα(G), respectively, as Aα(G) = αD(G)+(1−α)A(G) and Lα(G) = αD(G)+(α − 1)A(G). In this paper, we obtain some relationships between the eigenvalues of these matrices for some families of graphs, a part of the Aα and Lα-spectrum of the spider graphs, and we display the Aα and Lα-characteristic polynomials when their set of vertices can be partitioned into subsets that induce regular subgraphs. Moreover, we determine some subfamilies of spider graphs that are cospectral with respect to these matrices.
Anderson Fernandes Novanta, Leonardo de Lima, and Carla Silva Oliveira
Elsevier BV
Abstract Let G a graph on n vertices. The signless Laplacian matrix of G, denoted by Q ( G ) , is defined as Q ( G ) = D ( G ) + A ( G ) , where A ( G ) is the adjacency matrix of G and D ( G ) is the diagonal matrix of the degrees of G. A graph G is said to be Q-integral if all eigenvalues of the matrix Q ( G ) are integers. In this paper, we characterize all Q-integral graphs among all connected graphs with at most two vertices of degree greater than or equal to three.
Ansderson Fernandes Novanta, Carla Silva Oliveira, and Leonardo de Lima
Universidad Catolica del Norte - Chile
Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.
Nair Abreu, Jorge Alencar, Andre Brondani, Leonardo de Lima, and Carla Oliveira
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora
Abstract 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 are determined by their spectrum.
L.S. de Lima, A. Mohammadian, and C.S. Oliveira
Elsevier BV
Abstract A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this article, we characterize the integral graphs with exactly one vertex of degree larger than two. The integral graphs with two vertices of degree larger than two are characterized providing that those two vertices are not adjacent to each other. These generalize some known results about trees.
Nair Maria Maia de Abreu, Claudia Marcela Justel, Lilian Markenzon, Carla Silva Oliveira, and Christina Fraga Esteves Maciel Waga
Elsevier BV
Abstract 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 subclass of these graphs have integer Laplacian eigenvalues determined by the cardinalities of their maximal cliques.
Rodrigo Grijó, Leonardo de Lima, Carla Oliveira, Guilherme Porto, and Vilmar Trevisan
Elsevier BV
Abstract For a simple graph G with n vertices, let μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n − 1 ( G ) ≥ μ n ( G ) = 0 be the Laplacian eigenvalues of G . We study Nordhaus–Gaddum type inequalities for μ 1 ( G ) and μ 2 ( G ) . We improve some existing results from the literature for μ 1 ( G ) and obtain new results for μ 2 ( G ) . We also propose a conjecture that μ 2 ( G ) + μ 2 ( G ¯ ) ≤ 2 n − 2 holds for any graph G and prove it is true when G or G ¯ is disconnected; when G is bipartite; for regular graphs and when G and G ¯ have diameter not equal to 2 .
André Ebling Brondani, Carla Silva Oliveira, Francisca Andrea Macedo França, and Leonardo de Lima
Elsevier BV
Abstract Let G be a connected graph of order n, A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the row-sums of A(G). In 2017, Nikiforov [Nikiforov, V., Merging the A- and Q-Spectral Theories, Applicable Analysis and Discrete Mathematics 11 (2017), pp. 81–107.] defined the convex linear combinations Aα(G) of A(G) and D(G) by A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , 0 ≤ α ≤ 1 . In this paper, we obtain a partial factorization of the Aα-characteristic polynomial of the firefly graph which explicitly gives some eigenvalues of the graph.
Nair Abreu, André Brondani, Lima de, and Carla Oliveira
National Library of Serbia
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, ?2(G), ?2(G) + ?2(G?) ? -1 + ? n2/2-n+1, when G is a graph with girth at least 5. Also, we show that the bound above is tight. Besides, we prove that this result holds for some classes of connected graphs such as trees, k-cyclic, regular bipartite and complete multipartite graphs. Based on these facts, we conjecture that our result holds to any graph.
L. S. de Lima, A. Mohammadian, and C. S. Oliveira
Informa UK Limited
The eigenvalues of a graph are those of its adjacency matrix. Recently, Cioabă, Haemers, Vermette and Wong characterized all connected non-bipartite graphs with all but two eigenvalues equal to 1 or . In this article, we will generalize their result by explicitly determining all connected non-bipartite graphs with all but two eigenvalues in the interval .
Carla Silva Oliveira and Leonardo de Lima
Elsevier BV
Let $G$ be a graph of order $n \\geq 3$ with sequence degree given as $d_{1}(G) \\geq ... \\geq d_{n}(G)$ and let $\\mu_1(G),..., \\mu_n(G)$ and $q_1(G), ..., q_{n}(G)$ be the Laplacian and signless Laplacian eigenvalues of $G$ arranged in non increasing order, respectively. Here, we consider the Grone's inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin. Multilin. Alg. 39 (1995) 133--136] $$ \\sum_{i=1}^{k} \\mu_{i}(G) \\geq \\sum_{i=1}^{k} d_{i}(G)+1$$ and prove that for $k=2$, the equality holds if and only if $G$ is the star graph $S_{n}.$ The signless Laplacian version of Grone's inequality is known to be true when $k=1.$ In this paper, we prove that it is also true for $k=2,$ that is, $$q_{1}(G)+q_{2}(G) \\geq d_1(G)+d_2(G)+1$$ with equality if and only if $G$ is the star $S_{n}$ or the complete graph $K_{3}.$ When $k \\geq 3$, we show a counterexample.
Bruno Amaro, Leonardo de Lima, Carla Oliveira, Carlile Lavor, and Nair Abreu
Elsevier BV
Abstract 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 q 1 ( G ) , … , q n ( G ) be the eigenvalues of Q ( G ) in non-increasing order and let T k ( G ) = ∑ i = 1 k q i ( G ) be the sum of the k largest signless Laplacian eigenvalues of G. In this paper, we obtain an upper bound to T k ( H ) , when H is the P 3 -join graph isomorphic to P 3 [ ( n − k − 1 ) K 1 , K k − 1 , K 2 ] for 3 ≤ k ≤ n − 2 . Also, we conjecture that T k ( G ) is bounded above by T k ( H ) for any G with n vertices.
Leonardo de Lima, Vladimir Nikiforov, and Carla Oliveira
Elsevier BV
Let q min ( G ) stand for the smallest eigenvalue of the signless Laplacian of a graph G of order n . This paper gives some results on the following extremal problem:How large can q min ( G ) be if G is a graph of order n , with no complete subgraph of order r + 1 ? It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on q min are obtained, thus extending previous work of Brandt for regular graphs.In addition, the spectra of the Laplacian and the signless Laplacian of blowups of graphs are calculated. Finally, using graph blowups, a general asymptotic result about the maximum q min is established.
Carla Oliveira, Leonado Lima, Paula Rama, and Paula Carvalho
University of Wyoming Libraries
Let G be a simple graph on n vertices and e(G) edges. Consider the signless Laplacian, Q(G) = D + A, where A is the adjacency matrix and D is the diagonal matrix of the vertices degree of G. Let q_1(G) and q_2(G) be the first and the second largest eigenvalues of Q(G), respectively, and denote by S_n^+ the star graph with an additional edge. It is proved that inequality q_1(G)+q_2(G) \\leq e(G)+3 is tighter for the graph S_n^+ among all firefly graphs and also tighter to S_n^+ than to the graphs K_k \\vee K_{nâk} recently presented by Ashraf, Omidi and Tayfeh-Rezaie. Also, it is conjectured that S_n^+ minimizes f(G) = e(G) â q_1(G) â q_2(G) among all graphs G on n vertices.
Joelma Ananias de Oliveira, Carla Silva Oliveira, Claudia Justel, and Nair Maria Maia de Abreu
FapUNIFESP (SciELO)
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 or the variance of the vertex degrees. Among them there is the index of the graph, a spectral parameter, which is given as a function of the maximum eigenvalue of its adjacency matrix. In this paper, we survey these invariants with highlight to their respective properties, especially those relative to extremal graphs. Finally, we determine the maximum values of those measures and characterize their extremal graphs in some special classes.
Leonardo Silva de Lima, Carla Silva Oliveira, Nair Maria Maia de Abreu, and Vladimir Nikiforov
Elsevier BV
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 Laplacian.
Steve Kirkland, Carla Silva Oliveira, and Claudia Marcela Justel
Elsevier BV
Abstract Suppose that G is an undirected graph, and that H is a spanning subgraph of G c whose edges induce a subgraph on p vertices. We consider the expression α ( G ∪ H ) - α ( G ) , where α denotes the algebraic connectivity. Specifically, we provide upper and lower bounds on α ( G ∪ H ) - α ( G ) in terms of p , and characterise the corresponding equality cases. We also discuss the density of the expression α ( G ∪ H ) - α ( G ) in the interval [ 0 , p ] . A bound on α ( G ∪ H ) - α ( G ) is provided in a special case, and several examples are considered.
Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, and Steve Kirkland
Elsevier BV
The spread s(M) of an n × n complex matrix M is s(M) = maxij|�i − �j|, where the maximum is taken over all pairs of eigenvalues of M, �i,1 ≤ i ≤ n, [9] and [11]. Based on this concept, Gregory et al. [7] determined some bounds for the spread of the adjacency matrix A(G) of a simple graph G and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix. The signless Laplacian matrix of a graph G, Q(G) = D(G)+A(G), where D(G) is the diagonal matrix of degrees of G and A(G) is its adjacency matrix, has been recently studied, [4], [5]. The main goal of this paper is to determine some bounds on s(Q(G)). We prove that, for any graph on n ≥ 5 vertices, 2 ≤ s(Q(G)) ≤ 2n − 4, and we characterize the equality cases in both bounds. Further, we prove that for any connected graph G with n ≥ 5 vertices, s(Q(G)) < 2n − 4. We conjecture that, for n ≥ 5, sQ(G) ≤ √ 4n 2 − 20n + 33 and that, in this case, the upper bound is attained if, and only if, G is a certain pathcomplete graph.