|
Bipartite Graphs
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] |
Biclique K 3 5 Bicolor
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biregular Graph
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph G=(U,V,E) for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y, then the graph is said to be (x,y)-biregular. Example Every complete bipartite graph K_ is (b,a)-biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular. Vertex counts An (x,y)-biregular graph G=(U,V,E) must satisfy the equation x, U, =y, V, . This follows from a simple double counting argument: the number of endpoints of edges in U is x, U, , the number of endpoints of edges in V is y, V, , and each edge contributes the same amount (one) to both numbers. Symmetry Every regular bipartite graph is also biregular. Every edge-transitive graph (disallowing graphs with isolated vertices) that is not also vertex-transitive must be biregular.. In particular every e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squaregraph
In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face. Related graph classes The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos. As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices ''u'', ''v'', and ''w'' there is a unique median vertex ''m''(''u'',''v'',''w'') that lies on shortest paths between each pair of the three vertices.. See for a discussion of planar median graphs more generally. As with median graphs more generally, squaregraphs are also partial cubes: their vertices can be labeled with binary strings such that the Hamming distance between strings is equal to the shortest path distance between vertices. The graph obtained from a squaregraph by making a vertex for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. Square grid graph A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four Color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubters remain. The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map). To dispel any remaining doubts about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Kempe
Sir Alfred Bray Kempe FRS (6 July 1849 – 21 April 1922) was a mathematician best known for his work on linkages and the four colour theorem. Biography Kempe was the son of the Rector of St James's Church, Piccadilly, the Rev. John Edward Kempe. He was educated at St Paul's School, London and then studied at Trinity College, Cambridge, where Arthur Cayley was one of his teachers. He graduated BA (22nd wrangler) in 1872. Despite his interest in mathematics he became a barrister, specialising in the ecclesiastical law. He was knighted in 1913, the same year he became the Chancellor for the Diocese of London. He was also Chancellor of the dioceses of Newcastle, Southwell, St Albans, Peterborough, Chichester, and Chelmsford. He received the honorary degree DCL from the University of Durham and he was elected a Bencher of the Inner Temple in 1909. In 1876 he published his article ''On a General Method of describing Plane Curves of the nth degree by Linkwork,'' which presented a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of 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] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even number of vertices * 2-regular * 2-ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In 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 conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spink & Son
Spink & Son (established 1666) are an auction and collectibles company known principally for their sales of coins, banknotes, stock and bond certificates and medals. They also deal in philatelic items, wine and spirits, and other collectible items. History John Spink founded a goldsmith's and pawnbroker's business near Lombard Street, London, in 1666.Our History. Spink & Son. Retrieved 10 July 2020. The caused a temporary relocation before Spink returned to the rebuilt Lombard Street. In 1770 the firm moved to 2 |
Dominating Set
In graph theory, a dominating set for a graph is a subset of its vertices, such that any vertex of is either in , or has a neighbor in . The domination number is the number of vertices in a smallest dominating set for . The dominating set problem concerns testing whether for a given graph and input ; it is a classical NP-complete decision problem in computational complexity theory. Therefore it is believed that there may be no efficient algorithm that can compute for all graphs . However, there are efficient approximation algorithms, as well as efficient exact algorithms for certain graph classes. Dominating sets are of practical interest in several areas. In wireless networking, dominating sets are used to find efficient routes within ad-hoc mobile networks. They have also been used in document summarization, and in designing secure systems for electrical grids. Formal definition Given an undirected graph , a subset of vertices D\subseteq V is called a dominating se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
