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Magic Graph
A magic graph is a graph whose edges are labelled by the first ''q'' positive integers, where ''q'' is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first ''q'' positive integers are called supermagic. A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant. There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common. Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and ... [...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 (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), 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'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. 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 natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4x4 Magic Square Hierarchy
Four-wheel drive, also called 4×4 ("four by four") or 4WD, refers to a two-axled vehicle drivetrain capable of providing torque to all of its wheels simultaneously. It may be full-time or on-demand, and is typically linked via a transfer case providing an additional output drive shaft and, in many instances, additional gear ranges. A four-wheel drive vehicle with torque supplied to both axles is described as "all-wheel drive" (AWD). However, "four-wheel drive" typically refers to a set of specific components and functions, and intended off-road application, which generally complies with modern use of the terminology. Definitions Four-wheel-drive systems were developed in many different markets and used in many different vehicle platforms. There is no universally accepted set of terminology that describes the various architectures and functions. The terms used by various manufacturers often reflect marketing rather than engineering considerations or significant technical di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
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 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerhard Ringel
Gerhard Ringel (October 28, 1919 in Kollnbrunn, Austria – June 24, 2008 in Santa Cruz, California) was a German mathematician. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture (now the Ringel–Youngs theorem), a mathematical problem closely linked with the four color theorem. Although born in Austria, Ringel was raised in Czechoslovakia and attended Charles University before being drafted into the Wehrmacht in 1940 (after Germany had taken control of much of what had been Czechoslovakia). After the war Ringel served for over four years in a Soviet prisoner of war camp. He earned his PhD from the University of Bonn in 1951 with a thesis written under the supervision of Emanuel Sperner and Ernst Peschl. Ringel started his academic career as professor at the Free University Berlin. In 1970 he left Germany due to bureaucratic consequences of the German student movement, and continued his career at the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alison Marr
Alison M. Marr (born 1980) is an American mathematician and mathematics educator. Her research concerns graph theory and graph labeling, and she is also an advocate of inquiry-based learning in mathematics. She works as a professor of mathematics and computer science at Southwestern University in Texas. Education and career Marr graduated from Murray State University in 2002, and earned a master's degree in mathematics at Texas A&M University in 2004. She completed her Ph.D. in 2007 at Southern Illinois University; her dissertation, ''Labelings of Directed Graphs'', was supervised by Walter D. Wallis. She has been a member of the mathematics faculty at Southwestern University since 2007. She was department chair for 2015–2018. Beyond mathematics, her teaching at Southwestern has included a freshman seminar on television game shows. Contributions Inquiry-based learning Marr is an advocate of inquiry-based learning in mathematics, a style of teaching through student resea ... [...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] |
