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Graph Distance
In the mathematics, mathematical field of graph theory, the distance between two vertex (graph theory), vertices in a Graph (discrete mathematics), graph is the number of edges in a shortest path problem, 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 (graph theory), path connecting the two vertices, i.e., if they belong to different component (graph theory), 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 Metric (mathematics)#Quasimetrics, quasi-metric, and it mig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph)
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically to mean a measurement of the amount of difference between two similar objects (such as statistical distance between probability distributions or edit distance between string (computer science), strings of text) or a degree of separation (as exemplified by distance (graph theory), distance between people in a social network). Most such notions of distance, both physical and metaphorical, are formalized in mathematics using the notion of a metric space. In the social sciences, distance can refer to a qualitative measurement of separation, such as social distance or psychological distance. Distances in physics and geometry The distance between physical locations can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triameter (graph Theory)
In Graph theory, graph theory, the triameter is a metric Graph property, invariant that generalizes the concept of a Diameter (graph theory), graph's diameter. It is defined as the maximum sum of pairwise Distance (graph theory), distances between any three Vertex (graph theory), vertices in a Connectivity (graph theory), connected graph G and is denoted by \mathop(G) = \max\, where V is the vertex set of G and d(u,v) is the length of the shortest Path (graph theory), path between Vertex (graph theory), vertices u and v. It extends the idea of the Diameter (graph theory), diameter, which captures the longest Path (graph theory), path between any two of its Vertex (graph theory), vertices. A triametral triple is a Set (mathematics), set of three Vertex (graph theory), vertices achieving \mathop(G). History The parameter of triameter is related to the channel Assignment problem, assignment problem—the problem of assigning Frequency, frequencies to the Transmitter, transmitter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Diameter Problem
In graph theory, the degree diameter problem is the problem of finding the largest possible graph (in terms of the size of its vertex set ) of diameter such that the largest degree of any of the vertices in is at most . The size of is bounded above by the Moore bound; for and , only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter and degree attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound. Formula Let n_ be the maximum possible number of vertices for a graph with degree at most ''d'' and diameter ''k''. Then n_\leq M_, where M_ is the Moore bound: :M_ = \begin1 + d\frac & \textd>2 \\ 2k+1 & \textd=2\end This bound is attained for very few graphs, thus the study moves to how close there exist graphs to the Moore bound. For asymptotic behaviour note that M_ = d^k + O(d^). Define the parameter \mu_k=\liminf_\frac. It is conjectured that \mu_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closeness (graph Theory)
In graph theory and network theory, network analysis, indicators of centrality assign numbers or rankings to vertex (graph theory), nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociology, sociological origin.Newman, M.E.J. 2010. ''Networks: An Introduction.'' Oxford, UK: Oxford University Press. Definition and characterization of centrality indices Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes. The word "importance" ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.Newman, M.E.J. 2010. ''Networks: An Introduction.'' Oxford, UK: Oxford University Press. Definition and characterization of centrality indices Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes. The word "importance" has a wide number of meanings, leading to many d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betweenness Centrality
In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices, that is, there exists at least one path such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. Betweenness centrality was devised as a general measure of centrality: it applies to a wide range of problems in network theory, including problems related to social networks, biology, transport and scientific cooperation. Although earlier authors have intuitively described centrality as based on betweenness, gave the first formal definition of betweenness centrality. Betweenness centrality finds wide application in network theory; it represents the degree to which nodes stand between each other. For example, in a telecommunications network, a node with higher b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resistance Distance
In graph theory, the resistance distance between two vertices of a simple, connected graph, , is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to , with each edge being replaced by a resistance of one ohm. It is a metric on graphs. Definition On a graph , the resistance distance between two vertices and is : \Omega_:=\Gamma_+\Gamma_-\Gamma_-\Gamma_, :where \Gamma = \left(L + \frac\Phi\right)^+, with denotes the Moore–Penrose inverse, the Laplacian matrix of , is the number of vertices in , and is the matrix containing all 1s. Properties of resistance distance If then . For an undirected graph :\Omega_=\Omega_=\Gamma_+\Gamma_-2\Gamma_ General sum rule For any -vertex simple connected graph and arbitrary matrix : :\sum_(LML)_\Omega_ = -2\operatorname(ML) From this generalized sum rule a number of relationships can be derived depending on the choice of . Two of note are; :\begin \sum_\Omega_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance Matrix
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the ''distance'' being used to define this matrix may or may not be a metric (mathematics), metric. If there are elements, this matrix will have size . In graph-theoretic applications, the elements are more often referred to as points, nodes or vertices. Non-metric distance matrix In general, a distance matrix is a weighted adjacency matrix of some graph. In a Network (mathematics), network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes (where the number of steps in the path is bounded). This distance function, while well defined, is not a metric. There need be no restrictions on the weights oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Matrix
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse but a common criterion is that the number of non-zero elements is roughly equal to the number of rows or columns. By contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements (e.g., ''m'' × ''n'' for an ''m'' × ''n'' matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions. For example, consider a line of balls connected by springs from one to the next: this is a sparse system, as only adjacent balls are coupled. By contrast, if the same line of balls were to have springs connecting each ball to all other balls, the system would correspond to a dense matrix. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
