Quasi-Random Graphs, Pseudo-Random Graphs and Pseudorandom Binary Sequences, I. (Quasi-Random Graphs)

In the last decades many results have been proved on pseudo-randomness of binary sequences. In this series our goal is to show that using many of these results one can also construct large families of quasi-random, pseudo-random and strongly pseudo-random graphs. Indeed, it will be proved that if the first row of the adjacency matrix of a circulant graph forms a binary sequence which possesses certain pseudorandom properties (and there are many large families of binary sequences known with these properties), then the graph is quasi-random, pseudo-random or strongly pseudo-random, respectively. In particular, here in Part I we will construct large families of quasi-random graphs along these lines. (In Parts II and III we will present and study constructions for pseudo-random and strongly pseudo-random graphs, respectively.)