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In order to better understand dynamical functions on amounts of natural andman-made complex systems, lots of researchers from a wide range of disciplines,covering statistic physics, mathematics, theoretical computer science, and soon, have spent much time in doing this intriguing study. In this paper, thediscussed popularly topic, how to construct reasonable graph-model and then toexplain many features of realistic networks using previously presentedtheoretical models, is still our main work. Compared with many pre-existingdeterministic graph-model in single evolution way, our new graph-model can beconstructed using three types of growth ways to meet preferential attachmentmechanism. Meanwhile several typical indices associated with network researchwill be reported. In addition, some interesting findings will be shown,including the first handshake between Fibonacci series and "pure" preferentialattachment mechanism, an obvious relationship connecting two well-known rules,power-law and Zipf-law, and a common but useful equation on the basis of bothspanning trees number and the number of spanning trees with maximum leaves.Based on these foregoing discussions, we can demonstrate that our graph-modelobeys 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.

Handshake between Fibonacci series and pure preferential attachment mechanism on a graph-model