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Random Regular Graph
A random ''r''-regular graph is a graph selected from \mathcal_, which denotes the probability space of all ''r''-regular graphs on n vertices, where 3 \le r 0 is a positive constant, and d is the least integer satisfying (r-1)^ \ge (2 + \epsilon)rn \ln n then, asymptotically almost surely, a random ''r''-regular graph has diameter at most ''d''. There is also a (more complex) lower bound on the diameter of ''r''-regular graphs, so that almost all ''r''-regular graphs (of the same size) have almost the same diameter. The distribution of the number of short cycles is also known: for fixed m \ge 3, let Y_3,Y_4,...Y_m be the number of cycles of lengths up to m. Then the Y_iare asymptotically independent Poisson random variables with means \lambda_i=\frac Algorithms for random regular graphs It is non-trivial to implement the random selection of ''r''-regular graphs efficiently and in an unbiased way, since most graphs are not regular. The ''pairing model'' (also ''configuration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by 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. 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'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Béla Bollobás
Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul Erdős from the age of 14. Early life and education As a student, he took part in the first three International Mathematical Olympiads, winning two gold medals. Paul Erdős invited Bollobás to lunch after hearing about his victories, and they kept in touch afterward. Bollobás's first publication was a joint publication with ErdősBollobás, Béla; Erdös, Paul, Über graphentheoretische Extremalprobleme (Extremal problems in graph theory), Mat. Lapok 13, 143-152 (1962) on extremal problems in graph theory, written when he was in high school in 1962. With Erdős's recommendation to Harold Davenport and a long struggle for permission from the Hungarian authorities, Bollobás was able to spend an undergraduate year in Cambridge, England. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Graph
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of ''typical'' graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, ''random graph'' refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a ''random graph''. Models A random graph is obtained by starting with a set of ''n'' isolated vertices and adding successive edges between them at random. The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connectivity (graph Theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more Connected component (graph theory), isolated subgraphs. It is closely related to the theory of flow network, network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertex (graph theory), vertices and are called connected if contains a Path (graph theory), path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A Graph (discrete mathematics), graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a Path (graph theory), path between every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter (graph Theory)
In graph theory, the diameter of a connected undirected graph is the farthest distance between any two of its vertices. That is, it is the diameter of a set for the set of vertices of the graph, and for the shortest-path distance in the graph. Diameter may be considered either for weighted or for unweighted graphs. Researchers have studied the problem of computing the diameter, both in arbitrary graphs and in special classes of graphs. The diameter of a disconnected graph may be defined to be infinite, or undefined. Graphs of low diameter The degree diameter problem seeks tight relations between the diameter, number of vertices, and degree of a graph. One way of formulating it is to ask for the largest graph with given bounds on its degree and diameter. For any fixed degree, this maximum size is exponential in diameter, with the base of the exponent depending on the degree. The girth of a graph, the length of its shortest cycle, can be at most 2k+1 for a graph of diameter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: * ''Edges without own identity'': The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes. * ''Edges with own identity'': Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges. A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two. For some authors, the terms ''pseudograph'' and ''multigraph'' are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops. Undirected multigraph (edges without ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brendan McKay (mathematician)
Brendan Damien McKay (born 26 October 1951) is an Australian computer scientist and mathematician. He is currently an emeritus professor in the Research School of Computer Science at the Australian National University (ANU). He has published extensively in combinatorics. Born in Melbourne, McKay received a Ph.D. in mathematics from the University of Melbourne in 1980, and was appointed assistant professor of computer science at Vanderbilt University in Nashville in the same year (1980–1983). His thesis, ''Topics in Computational Graph Theory'', was written under the direction of Derek Holton. He was awarded the Australian Mathematical Society Medal in 1990. He was elected a Fellow of the Australian Academy of Science in 1997, and appointed professor of computer science at the ANU in 2000. Mathematics McKay is the author of at least 127 refereed articles. One of McKay's main contributions has been a practical algorithm for the graph isomorphism problem and its software imple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Graphs
In mathematics, random graph is the general term to refer to probability distributions over Graph (discrete mathematics), graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of ''typical'' graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, ''random graph'' refers almost exclusively to the Erdős–Rényi model, Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a ''random graph''. Models A random graph is obtained by starting with a set of ''n'' isolated vertices and add ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
