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Core (graph Theory)
In the mathematical field of graph theory, a core is a notion that describes behavior of a graph with respect to graph homomorphisms. Definition Graph C is a core if every homomorphism f:C \to C is an isomorphism, that is it is a bijection of vertices of C. A core of a graph G is a graph C such that # There exists a homomorphism from G to C, # there exists a homomorphism from C to G, and # C is minimal with this property. Two graphs are said to be homomorphism equivalent or hom-equivalent if they have isomorphic cores. Examples * Any complete graph is a core. * A cycle of odd length is a core. * A graph G is a core if and only if the core of G is equal to G. * Every two cycles of even length, and more generally every two bipartite graphs are hom-equivalent. The core of each of these graphs is the two-vertex complete graph ''K''2. * By the Beckman–Quarles theorem, the infinite unit distance graph on all points of the Euclidean plane or of any higher-dimensional Euclidean s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon Royle
Gordon F. Royle is a professor at the School of Mathematics and Statistics at The University of Western Australia. Royle is the co-author (with Chris Godsil) of the book ''Algebraic Graph Theory'' (Springer Verlag, 2001, ). Royle is also known for his research into the mathematics of Sudoku and his search for the Sudoku puzzle with the smallest number of entries that has a unique solution. Royle earned his Ph.D. in 1987 from the University of Western Australia under the supervision of Cheryl Praeger Cheryl Elisabeth Praeger (born 7 September 1948, Toowoomba, Queensland) is an Australian mathematician. Praeger received BSc (1969) and MSc degrees from the University of Queensland (1974), and a doctorate from the University of Oxford in 197 ... and Brendan McKay. References {{DEFAULTSORT:Royle, Gordon Living people Australian mathematicians Graph theorists University of Western Australia alumni University of Western Australia faculty Year of birth missing (livi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chris Godsil
Christopher David Godsil is a professor and the former Chair at the Department of Combinatorics and mathematical optimization, Optimization in the University of Waterloo Faculty of Mathematics, faculty of mathematics at the University of Waterloo. He wrote the popular textbook on algebraic graph theory, entitled ''Algebraic Graph Theory'', with Gordon Royle, His earlier textbook on algebraic combinatorics discussed distance-regular graphs and association schemes. Background He started the Journal of Algebraic Combinatorics, and was the Editor-in-Chief of the Electronic Journal of Combinatorics from 2004 to 2008. He is also on the editorial board of the Journal of Combinatorial Theory Series B and Combinatorica. He obtained his Ph.D. in 1979 at the University of Melbourne under the supervision of Derek Alan Holton. He wrote a paper with Paul Erdős, so making his Erdős number equal to 1.Paul Erdős, Chris D. Godsil, S. G. Krantz, and Torren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphic Equivalence
In the mathematics of graph theory, two graphs, ''G'' and ''H'', are called homomorphically equivalent if there exists a graph homomorphism f\colon G\to H and a graph homomorphism g\colon H\to G. An example usage of this notion is that any two cores of a graph are homomorphically equivalent. Homomorphic equivalence also comes up in the theory of databases. Given a database schema, two instances I and J on it are called homomorphically equivalent if there exists an instance homomorphism f\colon I\to J and an instance homomorphism g\colon J\to I. In fact for any category ''C'', one can define homomorphic equivalence. It is used in the theory of accessible categories The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates i ..., where "weak universality" is the best one can hope for in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subset V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. *Induced paths are induced subgraphs that are paths. The shortest path between ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Distance Graph
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on n vertices can have. The best known lower bound is slightly above linear in n—far from the upper bound, proportional to n^. The number of colors required to color unit distance graphs is also unknown (t ... [...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] |
Beckman–Quarles Theorem
In geometry, the Beckman–Quarles theorem, named after Frank S. Beckman and Donald A. Quarles Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. Beckman and Quarles published this result in 1953; it was later rediscovered by other and re-proved in multiple Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known. Statement and proof idea Formally, the result is as follows. Let f be a function or multivalued function from a d-dimensional Euclidean space to itself, and suppose that, for every pair of points p and q that are at unit distance from each other, every pair of images f(p) and f(q) are also at unit distance from each other. Then f must be an isometry: it is a one-to-one function that p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
