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Lévy Family Of Graphs
In graph theory, a branch of mathematics, a Lévy family of graphs is a family of graphs ''G''''n'', ''n'' = 1, 2, 3, ..., which possess a certain type of "compactness" or "tangledness". Many naturally occurring families of graphs are Lévy families. Many mathematicians have noted this fact and have expressed surprise that it does not appear to have a ready explanation. Formally, a family of graphs ''Gn'', ''n'' = 1, 2, 3, ..., is a Lévy family if, for any \varepsilon>0 : \lim_\alpha\left(G_n,\varepsilon\right) =0 where : \alpha(G,\varepsilon) = \max \left\. Here ''D'' is the graph diameter of ''G'', and ''A''(''n'') is the ''n''- graph neighborhood of ''A''. Note that the maximization ranges over subsets ''A'' of ''G'', subject to ''A'' being over half the size of ''G'' In words, this means that one can take a subset of size at least half of ''G'', and blow it up by only \epsilon of the graph diameter, and end up with n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Diameter
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] |
Graph Neighborhood
Graph may refer to: Mathematics * Graph (discrete mathematics), a structure made of vertices and edges ** Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discrete mathematics *Graph of a function *Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing *Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility * Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also * Complex network * Graf * Graff (other) * Graph database * Grapheme, in linguistics * Graphemics * Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") * List of information graphics software *Statistical graphics Statistical graphics, also known as statistical graphical techniques, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
