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中文摘要：
主要研究一种特殊的模糊图（即one-step模糊图）的性质。提出了one-step模糊图、Hamiltonian模糊图、r-正则模糊图、二部模糊图、连通模糊图等概念,给出了强one-step Hamiltonian模糊图、强one-step r-正则模糊图、强one-step二部模糊图、强one-step连通模糊图的构造、强one-step模糊图在笛卡尔积、合成、补运算下的的简易表达式、one-step模糊图的分解定理以及强one-step模糊图在笛卡尔积运算下保持不变的一些性质,证明了任意模糊图可以分解为one-step模糊图。
英文摘要：
In this paper, properties of a special kind of fuzzy graphs, called one-step fuzzy graph, are studied. The notions of one-step fuzzy graph, Hamiltonian fuzzy graph, r-regular fuzzy graph, bipartite fuzzy graph and connected fuzzy graph are defined. Structures of strong one-step Hamiltonian fuzzy graphs, strong one-step r-regular fuzzy graphs, strong one-step bipartite fuzzy graphs and strong one-step connected fuzzy graphs are obtained. Simpler expressions of operations of cartesian product, composition, and complete of strong one-step fuzzy graphs are given. Decomposition theorem of one-step fuzzy graphs and some properties of strong one-step fuzzy graphs which preserved under cartesian product are also proved. It is proved that every fuzzy graph can be decomposed into one-step fuzzy graphs.
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