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Friendship Graph
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph. By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship graphs are generalized by the triangular cactus graphs. Friendship theorem The friendship theorem of states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Graph 8
Friendship is a relationship of mutual affection between people. It is a stronger form of interpersonal bond than an "acquaintance" or an "association", such as a classmate, neighbor, coworker, or colleague. Although there are many forms of friendship, certain features are common to many such bonds, such as choosing to be with one another, enjoying time spent together, and being able to engage in a positive and supportive role to one another. Sometimes friends are distinguished from family, as in the saying "friends and family", and sometimes from lovers (e.g., "lovers and friends"), although the line is blurred with friends with benefits. Similarly, being in the ''friend zone'' describes someone who is restricted from rising from the status of friend to that of lover (see also unrequited love). Friendship has been studied in academic fields, such as communication, sociology, social psychology, anthropology, and philosophy. Various academic theories of friendship have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Cactus Graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cacti) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle. Properties Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge. The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph ''K''4. Triangular cactus A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free Graph
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding or triangle detection problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with m edges is triangle-free in time \tilde O\bigl(m^\bigr) where the \tilde O hides sub-polynomial factors. Here \omega is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán's Theorem
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a clique (graph theory), complete subgraph of a given size. It is one of the central results of extremal graph theory, an area studying the largest or smallest graphs with given properties, and is a special case of the forbidden subgraph problem on the maximum number of edges in a graph that does not have a given subgraph. An example of an n-vertex (graph theory), vertex graph that does not contain any (r+1)-vertex clique K_ may be formed by partitioning the set of n vertices into r parts of equal or nearly equal size, and connecting two vertices by an edge whenever they belong to two different parts. The resulting graph is the Turán graph T(n,r). Turán's theorem states that the Turán graph has the largest number of edges among all -free -vertex graphs. Turán's theorem, and the Turán graphs giving its extreme case, were first described and studied by Hun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Graph Theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various Graph property, graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graceful Labeling
In graph theory, a graceful labeling of a Graph (discrete mathematics), graph with edges is a graph labeling, labeling of its Vertex (graph theory), vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and inclusive.Virginia Vassilevska Williams, Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001PostScript/ref> A graph which admits a graceful labeling is called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major open problem in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC (not to be confused with Kotzig's conjecture on regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge-graceful Labeling
In graph theory, an edge-graceful labeling is a type of graph labeling for simple, connected graphs in which no two distinct edges connect the same two distinct vertices and no edge connects a vertex to itself. Edge-graceful labelings were first introduced by Sheng-Ping Lo in his seminal paper. Definition Given a graph , we denote the set of its edges by and that of its vertices by . Let be the cardinality of and be that of . Once a labeling of the edges is given, a vertex of the graph is labeled by the sum of the labels of the edges incident to it, modulo . Or, in symbols, the induced labeling on a vertex is given by :V(u)=\Sigma E(e) \mod , V(G), where is the resulting value for the vertex and is the existing value of an edge incident to . The problem is to find a labeling for the edges such that all the labels from to are used once and that the induced labels on the vertices run from to . In other words, the resulting set of labels for the edges should be , ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Polynomial
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics. History George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem. If P(G, k) denotes the number of proper colorings of ''G'' with ''k'' colors then one could establish the four color theorem by showing P(G, 4)>0 for all planar graphs ''G''. In this way he hoped to apply the powerful tools of analysis and algebra for studying the roots of polynomials to the combinatorial coloring problem. Hassler Whitney generalised Birkhoff’s polynomial from the planar case to general graphs in 1932. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
In graph theory, a proper edge coloring of a Graph (discrete mathematics), graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum Degree (graph theory), degree or . For some graphs, such as bip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an '' edge coloring'' assigns a color to each edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or region) so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metamath
Metamath is a formal language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and mathematical analysis, analysis, among others. By 2023, Metamath had been used to prove 74 of the 100 theorems of the "Formalizing 100 Theorems" challenge. At least 19 proof verifiers use the Metamath format. The Metamath website provides a database of formalized theorems which can be browsed interactively. Metamath language The Metamath language is a metalanguage for formal systems. The Metamath language has no specific logic embedded in it. Instead, it can be regarded as a way to prove that inference rules (asserted as axioms or proven later) can be applied. The largest database of proved theorems follows conventional first-order logic and ZFC set theory. The Metamath language design (emplo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig Huneke
Craig Lee Huneke (born August 27, 1951) is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia. Huneke graduated from Oberlin College with a bachelor's degree in 1973 and in 1978 earned a Ph.D. from the Yale University under Nathan Jacobson and David Eisenbud (''Determinantal ideal and questions related to factoriality''). As a post-doctoral fellow, he was at the University of Michigan. In 1979 he became an assistant professor and was at the Massachusetts Institute of Technology and the University of Bonn (1980). In 1981 he became an assistant professor at Purdue University, where in 1984 he became an associate professor and became a professor in 1987. From 1994 to 1995 he was a visiting professor at the University of Michigan and in 1999 was at the Max Planck Institute for Mathematics in Bonn (as a Fulbright Scholar). In 1999, he was Henry J. Bischoff professor at the University of Kansas. In 2002 he was at MSRI. Since 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
