Cografos com intervalos livres de autovalores

• Ano de defesa: 2024
• Autor(a) principal: Pedroso, Rafael Muhd
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de Santa Maria; Brasil; Matemática; UFSM; Programa de Pós-Graduação em Matemática; Centro de Ciências Naturais e Exatas
• Palavras-chave em Português: Cografos; Matriz de adjacência; Intervalos livres de autovalores; Cographs; Adjacency matrix; Eigenvalue-free interval; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.ufsm.br/handle/1/31954

The search for distribution of eigenvalues of a graph on the real line is a topic of interest in Spectral Graph Theory. Taking into account that any interval of the real line contains some eigenvalues of a graph, since any root of a real-root monic polynomial with integer coefficients occurs as an eigenvalue of some tree, however, the class of cographs has eigenvalues free interval, that is, eigenvalues that do not belong to a specific given interval. From this, and motivated by the structural and spectral characteristics of these graphs, and with the aid of Diagonalization Algorithm we show that the eigenvalues of a cograph are free from the interval Ω = (−1, 0). Posteriorly, using second-order Chebyshev polynomials and Toeplitz matrices, we refine the interval to Ω = [−1−√2 2 , −1+√2 2 ], proving to be valid for any threshold graph, a subclass of cographs. We also present in this dissertation two algorithms that generate sequences of threshold graphs with eigenvalues-free from the intervals (�������,−1) and (0,�������), where ������� and ������� are real numbers given such that ������� < −1 and ������� > 0. And finally, we present certain classes of cographs that have eigenvalues-free of the interval Ω = [−1−√2 2 , −1+√2 2 ����������������������������], where ���������������������������� is the smallest natural number of a given sequence.