Produto funcional de grafos :propriedades e aplicações

Abstract

In this thesis, we approach the functional product of graphs, their properties, and applications. We study associativity, and show that, in the set of graphs, the relation “to have the same sequence of degrees” is an equivalence relation and the functional product is associative in the equivalence classes. With regard to the invariant connectivity, we present conditions in which the functional product of bipartite graphs generates a disconnected graph, and we generalize this result to k-partite graphs. Therefore, we improved the result reported by Lozano et al. [33], which ensures the connectivity of the functional product graph, when the factor graphs are connected. Together with Lozano and Siqueira, we show that the functional product of graphs allows to construct harmonic graphs from any regular graph. Initially, we prove that for every regular graph G and its complement G′ , there are linking applications such that the functional product graph is harmonic. After that, we show that given a regular graph G and its complement G′ , if ∆(G′ ) is even, then for any graph H such that ∆(G′ ) = ∆(H), there are linking applications such that the functional product graph is harmonic. Finally, we prove that for n and k ∈ N, if (k + 1)|n, there is a harmonic connected k-regular graph with n vertices.