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Complex networks have seen much interest from all research fields and have found many potential applications in a variety of areas including natural, social, biological, and engineering technology. The deterministic models for complex networks play an indispensable role in the field of network model. The construction of a network model in a deterministic way not only has important theoretical significance, but also has potential application value. In this paper, we present a class of 3-regular network model with small world phenomenon. We determine its relevant topological characteristics, such as diameter and clustering coefficient. We also give a calculation method of number of spanning trees in the 3-regular network and derive the number and entropy of spanning trees, respectively.