74 research outputs found
A Near Cyclic (m1, m2, ..., mr )-Cycle System Of Complete Multigraph
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
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
Decompositions of complete graphs into cycles of arbitrary lengths
We show that the complete graph on vertices can be decomposed into
cycles of specified lengths if and only if is odd, for , and . We also show
that the complete graph on vertices can be decomposed into a perfect
matching and cycles of specified lengths if and only if
is even, for , and .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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