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Mixed Graph
In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, is an edge with an orientation and can be denoted as \overrightarrow or (u,v) (note that u is the tail and v is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as uv or ,v/math>. For the purpose of our application example we will not be considering loops or multiple edges of mixed graphs. A walk in a mixed graph is a sequence v_0,c_1,v_1,c_2,v_2,\dots,c_k,v_k of vertices and edges/arcs such that for all indices i, either c_i=v_v_ is an edge of the graph or c_i=\overrightarrow is an arc of the graph. This walk is a path if it does not repeat any edges, arcs, or vertices, except possibly the first and last vertices. A path is closed if its first and last vertices are the same, a ... [...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 wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Polynomial
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics. History George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem. If P(G, k) denotes the number of proper colorings of ''G'' with ''k'' colors then one could establish the four color theorem by showing P(G, 4)>0 for all planar graphs ''G''. In this way he hoped to apply the powerful tools of analysis and algebra for studying the roots of polynomials to the combinatorial coloring problem. Hassler Whitney generalised Birkhoff’s polynomial from the planar case to general graphs in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Rationale Citati ..... References {{reflist Publications established in 1985 Combinatorics journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moral Graph
In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models. The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non-adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph is an undirected graph in which each node of the original is now connected to its Markov blanket. The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be ''married'' by sharing an edge. Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Inference
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphical Model
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning. Types of graphical models Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce. Undirected Graphical Model The un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Graph
In the mathematical modeling of job shop scheduling problems, disjunctive graphs are a way of modeling a system of tasks to be scheduled and timing constraints that must be respected by the schedule. They are mixed graphs, in which vertices (representing tasks to be performed) may be connected by both directed and undirected edges (representing timing constraints between tasks). The two types of edges represent constraints of two different types: *If one task ''x'' must be performed earlier than a second task ''y'', this constraint is represented by a directed edge from ''x'' to ''y''. *If, on the other hand, two tasks ''x'' and ''y'' can be performed in either order, but not simultaneously (perhaps because they both demand the use of the same equipment or other resource), this non-simultaneity constraint is represented by an undirected edge connecting ''x'' and ''y''. Pairs of tasks that have no constraint on their ordering – they can be performed in either order or even simultane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Job Shop Scheduling
Job-shop scheduling, the job-shop problem (JSP) or job-shop scheduling problem (JSSP) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job scheduling problem, we are given ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' of varying processing times, which need to be scheduled on ''m'' machines with varying processing power, while trying to minimize the makespan – the total length of the schedule (that is, when all the jobs have finished processing). In the specific variant known as ''job-shop scheduling'', each job consists of a set of ''operations'' ''O''1, ''O''2, ..., ''On'' which need to be processed in a specific order (known as ''precedence constraints''). Each operation has a ''specific machine'' that it needs to be processed on and only one operation in a job can be processed at a given time. A common relaxation is the flexible job shop, where each operation can be processed on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Polynomial
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G and contains information about how the graph is connected. It is denoted by T_G. The importance of this polynomial stems from the information it contains about G. Though originally studied in algebraic graph theory as a generalization of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specializations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science. The Tutte polynomial has several equivalent definitions. It is equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin–Kasteley ... [...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 of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' 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'' means that ''A'' owes money to ''B'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a '' tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
