New Bounds on Graph Energy via Topological Indices Akbar Jahanbani, Carla Silva Oliveira, João da Silva Junior Domingos Gomes Match, 2026 This paper establishes a comprehensive framework of novel upper bounds for the energy of graphs, rigorously connecting this spectral invariant to a suite of topological indices. We derive a unified collection of theorems that express graph energy in terms of fundamental parameters-order, size, and extreme degrees-while intricately incorporating advanced indices such as the general zeroth-order Randić index, the Sombor index, and the atom-bond connectivity index. Our results not only generalize but also systematically refine many classical bounds in the literature, demonstrating the profound interplay between spectral graph theory and chemical graph theory.
Spectral heuristics applied to vertex reliability Carla Silva Oliveira, Fausto Pinheiro Junior, José André de Moura Brito Opsearch, 2026 The graph reliability is the probability that a connected graph remains connected after the removal of a random number of its vertices or edges. In this article, the problem addressed is identifying the topological change in a graph that leads to the greatest increase in its graph reliability. We restrict our domain to a subproblem consisting of the case in which removal occurs only on the vertices of a graph (vertex reliability), and the only topological change allowed is a single edge insertion. In this setting, we describe 6 heuristics, all previously used in the context of edge reliability or other robustness measures. Specifically, we further analyze two spectral heuristics and their theoretical motivations with respect to vertex reliability. The performance of these 6 heuristics is evaluated by a set of computational experiments with 22000 graphs of orders 10 up to 20, generated using the Erdős-Rényi, Barabási-Albert, and Watts-Strogatz models, that compared the vertex reliability of each edge insertion produced by the heuristics. From the experiments, one spectral heuristic presented a superior performance versus the others. We propose an explanation for why this spectral heuristic performed so well and how its underlying principle – the Fiedler vector – is intrinsically linked to local and global connectedness information of the graph. Additionally, we present an initial refinement of this spectral heuristic and compare it against the others in order to show potential developments using spectral measures.
Greedy recursive spectral bisection for modularity-bound hierarchical divisive community detection Douglas O. Cardoso, João Domingos Gomes da Silva Junior, Carla Silva Oliveira, Celso Marques, Laura Silva de Assis Statistics and Computing, 2024 Spectral clustering techniques depend on the eigenstructure of a similarity matrix to assign data points to clusters, so that points within the same cluster exhibit high similarity and are compared to those in different clusters. This work aimed to develop a spectral method that could be compared to clustering algorithms that represent the current state of the art. This investigation conceived a novel spectral clustering method, as well as five policies that guide its execution, based on spectral graph theory and embodying hierarchical clustering principles. Computational experiments comparing the proposed method with six state-of-the-art algorithms were undertaken in this study to evaluate the clustering methods under scrutiny. The assessment was performed using two evaluation metrics, specifically the adjusted Rand index, and modularity. The obtained results furnish compelling evidence, indicating that the proposed method is competitive and possesses distinctive properties compared to those elucidated in the existing literature. This suggests that our approach stands as a viable alternative, offering a robust choice within the spectrum of available same-purpose tools.
Some results involving the Aα-eigenvalues for graphs and line graphs João Domingos G. da Silva Júnior, Carla Silva Oliveira, Liliana Manuela G. C. da Costa Special Matrices, 2024 Let G G be a simple graph with adjacency matrix A ( G ) A\\left(G) , degree diagonal matrix D ( G ) , D\\left(G), and let l ( G ) l\\left(G) be the line graph of G G . In 2017, Nikiforov defined the A α {A}_{\\alpha } -matrix of G G , A α ( G ) {A}_{\\alpha }\\left(G) , as a linear convex combination of A ( G ) A\\left(G) and D ( G ) D\\left(G) , in the following way, A α ( G ) ≔ α A ( G ) + ( 1 − α ) D ( G ) , {A}_{\\alpha }\\left(G):= \\alpha A\\left(G)+\\left(1-\\alpha )D\\left(G), where α ∈ [ 0 , 1 ] \\alpha \\in \\left[0,1] . In this study, we present some bounds for the largest eigenvalue of A α ( G ) , {A}_{\\alpha }\\left(G), and for some eigenvalues of A α ( l ( G ) ) {A}_{\\alpha }\\left(l\\left(G)) . Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.
Bounding the sum of the largest signless Laplacian eigenvalues of a graph Aida Abiad, Leonardo de Lima, Sina Kalantarzadeh, Mona Mohammadi, Carla Oliveira Discrete Applied Mathematics, 2023 We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
Bounds for Aα-eigenvalues João Domingos Gomes da Silva, Carla Silva Oliveira, Liliana Manuela G.C. da Costa RAIRO Operations Research, 2023 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.
A Survey on Graovac-Ghorbani Index Diego Pacheco, , Carla Oliveira, Anderson Novanta, , and Match, 2023 Let G = (V, E) be a simple undirected and connected graph on n vertices.The Graovac-Ghorbani (ABCGG) index of a graph G is defined aswhere n(u) is the number of vertices closer to vertex u than vertex v and n(v) is defined analogously.This paper is a survey of topological Graovac-Ghorbani index of a graph G.It contains results on ABCGG which are known until this moment and some conjectures.
TSPredIT: Integrated Tuning of Data Preprocessing and Time Series Prediction Models Rebecca Salles, Esther Pacitti, Eduardo Bezerra, Celso Marques, Carla Pacheco, Carla Oliveira, Fabio Porto, Eduardo Ogasawara Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics, 2023
Positive semidefiniteness of Aα(G) on some families of graphs A.E. Brondani, F.A.M. França, C.S. Oliveira Discrete Applied Mathematics, 2022 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.
Aα-Spectrum of a Firefly Graph André Ebling Brondani, Carla Silva Oliveira, Francisca Andrea Macedo França, Leonardo de Lima Electronic Notes in Theoretical Computer Science, 2019
Measures of irregularity of graphs Joelma Ananias de Oliveira, Carla Silva Oliveira, Claudia Justel, Nair Maria Maia de Abreu Pesquisa Operacional, 2013
Bounds on the Q-spread of a graph Carla Silva Oliveira, Leonardo Silva de Lima, Nair Maria Maia de Abreu, Steve Kirkland Linear Algebra and Its Applications, 2010
Laplacian integral graphs in S(a, b) Leonardo Silva de Lima, Nair Maria Maia de Abreu, Carla Silva Oliveira, Maria Aguieiras Alvarez de Freitas Linear Algebra and Its Applications, 2007
