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Kautz Graph
The Kautz graph K_M^ is a directed graph of degree M and dimension N+ 1, which has (M +1)M^ vertices labeled by all possible strings s_0 \cdots s_N of length N + 1 which are composed of characters s_i chosen from an alphabet A containing M + 1 distinct symbols, subject to the condition that adjacent characters in the string cannot be equal (s_i \neq s_). The Kautz graph K_M^ has (M + 1)M^ edges \ \, It is natural to label each such edge of K_M^ as s_0s_1 \cdots s_, giving a one-to-one correspondence between edges of the Kautz graph K_M^ and vertices of the Kautz graph K_M^. Kautz graphs are closely related to De Bruijn graphs. Properties * For a fixed degree M and number of vertices V = (M + 1) M^N, the Kautz graph has the smallest diameter of any possible directed graph with V vertices and degree M. * All Kautz graphs have Eulerian cycles. (An Eulerian cycle is one which visits each edge exactly once—This result follows because Kautz graphs have in-degree equal to ou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kautz is a surname. Notable people with the surname include: *Albert Kautz (1839–1907), American naval officer, born at Georgetown, Ohio *August Kautz (1828–1895), German-American soldier and Union Army cavalry officer during the American Civil War *Christian Kautz (1913–1948), auto racing driver from Switzerland *Jacob Kautz (1500–1532), Anabaptist who posted seven theses to the door of the Worms Cathedral in 1527 * Julius Kautz (1829–1909), Hungarian economist See also * Kautz Creek, tributary of the Nisqually River in the Mount Rainier National Park of Washington * Kautz Creek Falls, waterfall on Kautz Creek in the Mount Rainier National Park of Washington *Kautz Glacier, narrow glacier on the southern flank of Mount Rainier in Washington *Kautz Family YMCA Archives collects the historical records of its national organization, the YMCA of the USA * Kautz filter, named after William H. Kautz, is a fixed-pole traversal filter, published in 1954 *Kautz graph The Kautz gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn Graph
In graph theory, an -dimensional De Bruijn graph of symbols is a directed graph representing overlaps between sequences of symbols. It has vertices, consisting of all possible sequences of the given symbols; the same symbol may appear multiple times in a sequence. For a set of symbols the set of vertices is: :V=S^n=\. If one of the vertices can be expressed as another vertex by shifting all its symbols by one place to the left and adding a new symbol at the end of this vertex, then the latter has a directed edge to the former vertex. Thus the set of arcs (that is, directed edges) is :E=\. Although De Bruijn graphs are named after Nicolaas Govert de Bruijn, they were discovered independently by both De Bruijn and I. J. Good. Much earlier, Camille Flye Sainte-Marie implicitly used their properties. Properties * If , then the condition for any two vertices forming an edge holds vacuously, and hence all the vertices are connected, forming a total of edges. * Each vertex has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph Theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with —so it is just a quasi-metric, and it might be the case that one is defined while the other is not. Related concepts A metric space defined over a set of points in terms of distances in a graph defined over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eulerian Cycle
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. The problem can be stated mathematically like this: :Given the graph in the image, is it possible to construct a path (or a cycle; i.e., a path starting and ending on the same vertex) that visits each edge exactly once? Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree, and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer. This is known as Euler's Theorem: :A connected grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Cycle
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Topology
Network topology is the arrangement of the elements ( links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks. Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network’s physical topology is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High-performance Computing
High-performance computing (HPC) uses supercomputers and computer clusters to solve advanced computation problems. Overview HPC integrates systems administration (including network and security knowledge) and parallel programming into a multidisciplinary field that combines digital electronics, computer architecture, system software, programming languages, algorithms and computational techniques. HPC technologies are the tools and systems used to implement and create high performance computing systems. Recently, HPC systems have shifted from supercomputing to computing clusters and grids. Because of the need of networking in clusters and grids, High Performance Computing Technologies are being promoted by the use of a collapsed network backbone, because the collapsed backbone architecture is simple to troubleshoot and upgrades can be applied to a single router as opposed to multiple ones. The term is most commonly associated with computing used for scientific research or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fault-tolerant Computing
Fault tolerance is the property that enables a system to continue operating properly in the event of the failure of one or more faults within some of its components. If its operating quality decreases at all, the decrease is proportional to the severity of the failure, as compared to a naively designed system, in which even a small failure can cause total breakdown. Fault tolerance is particularly sought after in high-availability, mission-critical, or even life-critical systems. The ability of maintaining functionality when portions of a system break down is referred to as graceful degradation. A fault-tolerant design enables a system to continue its intended operation, possibly at a reduced level, rather than failing completely, when some part of the system fails. The term is most commonly used to describe computer systems designed to continue more or less fully operational with, perhaps, a reduction in throughput or an increase in response time in the event of some partial f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Families Of Graphs
Parametric may refer to: Mathematics *Parametric equation, a representation of a curve through equations, as functions of a variable *Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribution * Parametric derivative, a type of derivative in calculus *Parametric model, a family of distributions that can be described using a finite number of parameters *Parametric oscillator, a harmonic oscillator whose parameters oscillate in time *Parametric surface, a particular type of surface in the Euclidean space R3 *Parametric family, a family of objects whose definitions depend on a set of parameters Science * Parametric process, in optical physics, any process in which an interaction between light and matter does not change the state of the material *Spontaneous parametric down-conversion, in quantum optics, a source of entangled photon pairs and of single photons *Optical parametric amplifier, a type of laser light source that em ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
