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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] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digraph Realization Problem
The digraph realization problem is a decision problem in graph theory. Given pairs of nonnegative integers ((a_1,b_1),\ldots,(a_n,b_n)), the problem asks whether there is a labeled directed graph, simple directed graph such that each vertex (graph theory), vertex v_i has directed graph, indegree a_i and directed graph, outdegree b_i. Solutions The problem belongs to the complexity class P (complexity), P. Two algorithms are known to prove that. The first approach is given by the Kleitman–Wang algorithms constructing a special solution with the use of a Recursion (computer science), recursive algorithm. The second one is a characterization by the Fulkerson–Chen–Anstee theorem, i.e. one has to validate the correctness of n inequalities. Other Notations The problem can also be stated in terms of zero-one matrix (mathematics), matrices. The connection can be seen if one realizes that each directed graph has an adjacency matrix where the column sums and row sums correspond to (a_ ... [...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] |
Erdős–Gallai Theorem
The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics. It provides one of two known approaches to solving the graph realization problem, i.e. it gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph. A sequence obeying these conditions is called "graphic". The theorem was published in 1960 by Paul Erdős and Tibor Gallai, after whom it is named. Statement A sequence of non-negative integers d_1\geq\cdots\geq d_n can be represented as the degree sequence of a finite simple graph on ''n'' vertices if and only if d_1+\cdots+d_n is even and : \sum^_d_i\leq k(k-1)+ \sum^n_ \min (d_i,k) holds for every k in 1\leq k\leq n. Proofs It is not difficult to show that the conditions of the Erdős–Gallai theorem are necessary for a sequence of numbers to be graphic. The requirement that the sum of the degrees be even is the handshaking lemma, already used by Euler in his 173 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices are Neighbourhood (graph theory), adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is Glossary of graph theory terms#undirected, undirected (i.e. all of its Glossary of graph theory terms#edge, edges are bidirectional), the adjacency matrix is symmetric matrix, symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are Incidence (graph), incident or not, and its degree matrix, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majorization
In mathematics, majorization is a preorder on vector space, vectors of real numbers. For two such vectors, \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below, commonly denoted \mathbf \succ_w \mathbf, when : \sum_^k x_i^ \geq \sum_^k y_i^ for all k=1,\,\dots,\,n, where x_i^ denotes ith largest entry of x. If \mathbf, \mathbf further satisfy \sum_^n x_i = \sum_^n y_i, we say that \mathbf majorizes (or dominates) \mathbf , commonly denoted \mathbf \succ \mathbf. Both weak majorization and majorization are partially ordered set, partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). Specifically, \mathbf \succ \mathbf \wedge \mathbf \succ \mathbf if and only if \mathbf, \mathbf are permutations of each other. Similarly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Counting (proof Technique)
In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which call "one of the most important tools in combinatorics", one describes a finite set from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. Examples Multiplication (of natural numbers) commutes This is a simple example of double counting, often used when teaching multiplication to young children. In this context, multiplication of natural numbers is introduced as repeated addition, and is then shown to be commutative by counting, in two different ways, a number of items arranged in a rectangular grid. Suppose the grid has n rows and m columns. We first count the items by summing n rows of m items each, then a second time by summing m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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_m) 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^m_ \min(b_i,k). Sometimes this theorem is stated with the additional constraint b_1\geq\cdots\geq b_m. This condition is not necessary, because the labels of vertices of one partite set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
