Deterministic self-similar models of complex networks based on very symmetric graphs

Using very symmetric graphs we generalize several deterministic self-similar models of complex networks and we calculate the main network parameters of our generalization. More specifically, we calculate the order, size and the degree distribution, and we give an upper bound for the diameter and a lower bound for the clustering coefficient. These results yield conditions under which the network is a self-similar and scale-free small world network. We remark that all these conditions are posed on a small base graph which is used in the construction. As a consequence, we can construct complex networks having prescribed properties. We demonstrate this fact on the clustering coefficient. We propose eight new infinite classes of complex networks. One of these new classes is so rich that it is parametrized by three independent parameters.