Cografos integrais

Abstract

The search for integral cographs is a topic of interest in Spectral Graph Theory. From this, and motivated by the structural characteristics of these graphs, and by their spectral properties, we propose, in this dissertation, to show that two distinct techniques (algorithmic and combinatorial) can be effectively used to characterize, or determine classes of integral cographs. Through balanced cotrees, we started from cographs associated with balanced cotrees, and with the aid of Algorithm Diagonalize(𝑇𝐺𝑟 , 𝑥) we determine the eigenvalues of the respective cograph, which are integers; and through combinatorial triangles, we determine which cographs are integrals, of the associates with the triangle Determinant Hosoya Triangle ℋ. The main results obtained are the Theorem 3.4.4, from the article by Allem and Tura (2020), the Theorem 4.3.8 and the Proposition 4.3.6, from the article by Ching, Flórez and Mukhrjee (2020). These lead us to characterize that for 𝑤 = 3𝑡 and 𝑤 = 3𝑡 + 1, the cographs, respectively, with and without loops, associated to the adjacency matrices 𝒮* 𝑤 𝑚𝑜𝑑 2 and 𝒮𝑤 𝑚𝑜𝑑 2, are integrals; and cographs with balanced cotrees 𝑇𝐺𝑟 , also they are integrals.