In order to better understand dynamical functions on amounts of natural and man-made complex systems, lots of researchers from a wide range of disciplines, covering statistic physics, mathematics, theoretical computer science, and so on, have spent much time in doing this intriguing study. In this paper, the discussed popularly topic, how to construct reasonable graph-model and then to explain many features of realistic networks using previously presented theoretical models, is still our main work. Compared with many pre-existing deterministic graph-model in single evolution way, our new graph-model can be constructed using three types of growth ways to meet preferential attachment mechanism. Meanwhile several typical indices associated with network research will be reported. In addition, some interesting findings will be shown, including the first handshake between Fibonacci series and "pure" preferential attachment mechanism, an obvious relationship connecting two well-known rules, power-law and Zipf-law, and a common but useful equation on the basis of both spanning trees number and the number of spanning trees with maximum leaves. Based on these foregoing discussions, we can demonstrate that our graph-model obeys power-law and small-world property. For the future research directions, we present some unknown problems to be studied at the end of this paper.