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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.

In recent years, research on complex networks is in the ascendant. Complex networks by using graph theory and some methods of statistical physics can be used to capture and describe the evolution of the system mechanism, evolution pattern, and the overall behavior, which is one of the main causes of vigorous development on complex networks research [

Construction of small-world networks that conform to the real system features not only has important theoretical significance but also has potential application value in deterministic way [

The degree

Construction of the deterministic 3-regular network

Now we compute the size and order of

Due to the determinacy, the relevant characteristics of our model described above can be solved exactly. In the following we concentrate on the diameter and clustering coefficient.

Small-world networks describe many real-life networks; that is, there is a relatively short distance between most pairs of nodes in most real-life networks and their average path length (APL) does not increase linearly with the system size but grows logarithmically with the number of nodes or slower. The average path length is the smallest number of links connecting a pair of nodes, averaged over all pairs of nodes [

So, the diameter

Clustering is another important property of a complex network, which provides a measure of the local structure within the network [

Based on the above discussion, we can conclude that our model is a deterministic small-world network, because it is sparse with small diameter and average path length and high clustering coefficient.

A spanning tree of any connected network is defined as a minimal set of edges that connect every node. The enumeration of spanning trees in networks is a fundamental issue in mathematics [

In order to calculate the number of spanning trees accurately, at first we build a new model

Construction of the deterministic 3-regular network

The network models

We calculate the number of spanning trees of

The number of spanning trees of

And the shape of the spanning forests with two components such as

The number of spanning forests with two components of

By considering the symmetric, it is not hard to get the recurrence relation of the number of spanning trees and spanning forests of

Noticing that

Note that

So, the entropy of the spanning trees of

We can compare the entropy of the spanning trees of

In this paper, we proposed a class of deterministic regular small-world model which is constructed in an iterative manner and presented an exhaustive analysis of many properties of considered model. Then we obtained the analytic solutions for most of the topological features, including diameter and clustering coefficient. We also determined the number of spanning trees in the 3-regular small-world network. In addition, using the algorithm, we obtained the entropies of spanning trees.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the National Science Foundation of China (nos. 61164005 and 60863006), the National Basic Research Program of China (no. 2010CB334708), and Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1068).