74 research outputs found

    Uniform cycle decompositions of complete multigraphs

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    A Near Cyclic (m1, m2, ..., mr )-Cycle System Of Complete Multigraph

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    Let v, λ be positive integers, λKv denote a complete multigraph on v vertices in which each pair of distinct vertices joining with λ edges. In this article, difference method is used to introduce a new design that decomposes 4Kv into cycles, when v ≡ 2, 10(mod12). This design merging between cyclic (m1, ..., mr ) -cycle system and near-fourfactor is called a near cyclic (m1, ..., mr ) -cycle syste

    Multilayer Networks

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    In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

    Computational Graph Theory

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    Decompositions of complete graphs into cycles of arbitrary lengths

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    We show that the complete graph on nn vertices can be decomposed into tt cycles of specified lengths m1,…,mtm_1,\ldots,m_t if and only if nn is odd, 3≤mi≤n3\leq m_i\leq n for i=1,…,ti=1,\ldots,t, and m1+⋯+mt=(n2)m_1+\cdots+m_t=\binom n2. We also show that the complete graph on nn vertices can be decomposed into a perfect matching and tt cycles of specified lengths m1,…,mtm_1,\ldots,m_t if and only if nn is even, 3≤mi≤n3\leq m_i\leq n for i=1,…,ti=1,\ldots,t, and m1+…+mt=(n2)−n2m_1+\ldots+m_t=\binom n2-\frac n2.Comment: 182 pages, 0 figures, A condensed version of this paper was published as "Cycle decompositions V: Complete graphs into cycles of arbitrary lengths" (see reference [24]). Here, we include supplementary data and some proofs which were omitted from that pape
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