摘　　要
雙迴圈網路在區域網路架構或是在分散式記憶體多處理器系統中是一種廣為使用的網路架構，因為它具有較小的直徑、容易路遊，而且在各個節點容易切換至另一個連結，因此它是一個非常穩定的網路架構。N表示網路中節點的個數，d(N)表示在一N個節點的雙迴圈網路中有可能的最短直徑，對於任一給定的N值， Bermond et al. [5], Boesch and Wang [7], 和 Yebra et al. [23]已經證明了 ，這是一個廣為人知的下界值，標記為lb(N)。此篇論文所討論的主題為給定一個N值，如何決定一個最佳的拓樸使得d(N)= lb(N)，我們提供了一個非常簡單的計算公式，可以快速地找出搭配N值的最佳拓樸，可使得雙迴圈網路達到最佳化。

Abstract
　　Double-loop networks have become one of the most popular architectures in the design of Local Area Networks and distributed memory multiprocessor systems. This is due to its characters of minimal diameter, easy routing, expandability and regularity. The switching mechanism at each node can easily be implemented using building blocks of the same specification. Therefore, double-loop networks have a high degree of reliability and hence very low vulnerability. Let N denote the number of nodes in a double-loop network and d(N) be the best possible diameter with N vertices. Given an N, Bermond et al. [5], Boesch and Wang [7], and Yebra et al. [23] have shown that . This is a well-known lower bound for d(N) and is usually denoted as lb(N). In this paper, what we discuss is finding an optimal topology such that d(N)= lb(N) for any given value of N. We provide a simple formula to find optimal topologies of double-loop networks with N nodes.

Contents Page
Abstract (in Chinese) …………………………………………………………… I
Abstract (in English) ……………………………………………………………. II
Contents …………………………………………………………………………. III
List of Figures ………………………………………………………………….. IV
List of Tables ………………………………………………………………….. V
1. Introduction …………………………………………………………………… 1
1.1 About this work …………………………………………………………. 2
2. Problem Definition and Notation …………………………………………. 4
3. Relevant Results …………………………………………………………….. 8
4. Main Results …………………………………………………………………. 13
4.1 Simple Formula to Compute Optimal N and h ……………………… 13
4.2 Comparison with Other Results ………………………………………. 18
5. Future Work …………………………………………………………………. 20
6. References ………………………………………………………………….. 21

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