|
Gale–Ryser Theorem
The Gale–Ryser theorem is a result in graph theory and combinatorial matrix theory, two branches of combinatorics. It provides one of two known approaches to solving the bipartite realization problem, i.e. it gives a necessary and sufficient condition for two finite sequences of natural numbers to be the degree sequence of a labeled simple bipartite graph; a sequence obeying these conditions is called "bigraphic". It is an analog of the Erdős–Gallai theorem for simple graphs. The theorem was published independently in 1957 by H. J. Ryser and David Gale. Statement A pair of sequences of nonnegative integers (a_1,\ldots,a_n) and (b_1,\ldots,b_n) with a_1\geq\cdots\geq a_n is bigraphic if and only if \sum_^a_i=\sum_^b_i and the following inequality holds for all k \in \: : \sum^k_ a_i\leq \sum^n_ \min(b_i,k). Sometimes this theorem is stated with the additional constraint b_1\geq\cdots\geq b_n. This condition is not necessary, because the labels of vertices of one partite s ... [...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] |
Majorization
In mathematics, majorization is a preorder on vectors of real numbers. Let ^_,\ i=1,\,\ldots,\,n denote the i-th largest element of the vector \mathbf\in\mathbb^n. Given \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below (or equivalently, we say that \mathbf is weakly majorized (or dominated) by \mathbf from below) denoted as \mathbf \succ_w \mathbf if \sum_^k x_^ \geq \sum_^k y_^ for all k=1,\,\dots,\,d. If in addition \sum_^d x_i^ = \sum_^d y_i^, we say that \mathbf majorizes (or dominates) \mathbf , written as \mathbf \succ \mathbf , or equivalently, we say that \mathbf is majorized (or dominated) by \mathbf. The order of the entries of the vectors \mathbf or \mathbf does not affect the majorization, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). As a consequence, majorization is not a partial order, since \mathbf \succ \mathbf and \mathbf \succ \mathbf do not imply \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fulkerson–Chen–Anstee Theorem
The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)) to be the indegree-outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. FulkersonD.R. Fulkerson: ''Zero-one matrices with zero trace.'' In: ''Pacific J. Math.'' No. 12, 1960, pp. 831–836 (1960) obtained a characterization analogous to the classical Erdős–Gallai theorem for graphs, but in contrast to this solution with exponentially many inequalities. In 1966 Chen Wai-Kai Chen: ''On the realization of a (''p'',''s'')-digraph with prescribed degrees .'' In: ''Journal of the Franklin Institute'' No. 6, 1966, pp. 406–422 improved this result in demanding the additional constraint that the integer pairs must be sorted in non-increasing lex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Threshold Graph
In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications of the following two operations: # Addition of a single isolated vertex to the graph. # Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices. For example, the graph of the figure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were first introduced by . A chapter on threshold graphs appears in , and the book is devoted to them. Alternative definitions An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real vertex weight w(v) such that for any two vertices v,u, uv is an edge if and only if w(u)+w(v)> S. Another equivalent definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Threshold Sequences
Threshold may refer to: Architecture * Threshold (door), the sill of a door Media * ''Threshold'' (1981 film) * ''Threshold'' (TV series), an American science fiction drama series produced during 2005-2006 * "Threshold" (''Stargate SG-1''), an episode of the TV series * "Threshold" (''Star Trek: Voyager''), an episode of the TV series * Threshold Entertainment, a Hollywood Intellectual Property Management and Production Company * Threshold Podcast, a podcast focused on long-form reporting of climate justice topics Literature * ''Threshold'' (1990 novel), a science fiction novel by Chris and Janet Morris * ''Threshold'' (Sara Douglass novel), a fantasy novel * ''Threshold'' (Palmer novel), a science fiction novel by David R. Palmer * ''Threshold'', the first volume of the collected short fiction of Roger Zelazny * ''Threshold'' (DC Comics), a comic book published by DC Comics * Threshold (''Doctor Who''), an organization in ''Doctor Who'' comic strips * ''Threshold'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (graph Theory)
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ... which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions For a commutative ring ''R'' and an ''R''-Module (mathematics), module ''M'', an element ''r'' in ''R'' is called a non-zero-divisor on ''M'' if ''r m'' = 0 implies ''m'' = 0 for ''m'' in ''M''. An ''M''-regular sequence is a sequence :''r''1, ..., ''r''''d'' in ''R'' such that ''r''''i'' is a not a zero-divisor on ''M''/(''r''1, ..., ''r''''i''-1)''M'' for ''i'' = 1, ..., ''d''. Some authors also require that ''M''/(''r''1, ..., ''r''''d'')''M'' is not zero. Intuitively, to say that '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferrers Diagram
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general. Examples The seven partitions of 5 are: * 5 * 4 + 1 * 3 + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop (graph Theory)
In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices): * Where graphs are defined so as to ''allow'' loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a ''simple graph''. * Where graphs are defined so as to ''disallow'' loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a ''multigraph'' or ''pseudograph''. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet. Degree For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special case i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
