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中文摘要：
人们在实践中发现,网络拓扑结构的一些性质能够在某种程度上衡量一个网络的性能如何,网络的可靠性便是其中的一个重要性能指标.分析现实世界中已有网络,如计算机网络、电网以及通讯网络等的可靠性具有重要的理论意义和应用价值.图的字典乘积利用已有规模较小的网络来构建规模较大的网络,且所得大网络的特征值完全由小网络的拓扑结构参数来刻画,并具有良好的性能,而图的欧拉回路与欧拉迹亦在此领域有着广泛的应用.乘积因子图的拓扑结构影响着字典乘积图的拓扑结构.本文主要研究字典乘积图的Euler回路问题和Euler迹问题,利用组合理论和极值构造方法,给出了两图的字典乘积图为Euler回路和Euler迹的一些充分必要条件.
英文摘要：
It is found in practice that some properties of the network topological structure can measure the function of a network,and the network reliability is one of the most important factors.Analyzing the reliability of real networks,like the computer network,electric network and the communication network,owns great theoretic implication and application value.Actually,using the method of lexicographic product can easily construct a large graph from some smaller graphs,and the eigenvalues of the large graph are completely described by the parameters of the topological structure of smaller graphs.The Euler graph and Euler trail have a broad application in this field.In this paper,we mainly consider whether a lexicographic product graph,whose topological structure is affected by the factor graphs,is Euler graph or Euler trail.By using the theory of combination and the way of extreme construction,some necessary and sufficient conditions are given to ensure the lexicographic product is an Euler graph or Euler trail.
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