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Frucht Graph
In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939. The Frucht graph is a pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Like every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex-connected) and Hamiltonian, with girth 3. Its independence number is 5. The Frucht graph can be constructed from the LCF notation: . Algebraic properties The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity (that is, every vertex can be distinguished topologically from every other vertex). Such graphs are called asymmetric (or identity) graphs. Frucht's theorem states that any group can be realized as the group of symmetries of a graph,. and a strengthening of this theorem also due to Frucht states that any group can be realized as the symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frucht Planar Lombardi
Frucht is surname of: * (1913, Torgau - 1993), German doctor and physiologist *Michael Frucht (1969-), an American neurologi * Robert Frucht, Robert ''(Roberto)'' Wertheimer Frucht (1906 - 1997), a German-Chilean mathematician :* Frucht graph :* Frucht's theorem See also * Frucht Quark * Frücht, a small municipality in the federal state of Rhineland-Palatinate in western Germany {{surname, Frucht, Frukht German-language surnames Jewish surnames Yiddish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Individual Graphs
An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of being an individual; particularly (in the case of humans) of being a person unique from other people and possessing one's own needs or goals, rights and responsibilities. The concept of an individual features in diverse fields, including biology, law, and philosophy. Etymology From the 15th century and earlier (and also today within the fields of statistics and metaphysics) ''individual'' meant " indivisible", typically describing any numerically singular thing, but sometimes meaning "a person". From the 17th century on, ''individual'' has indicated separateness, as in individualism. Law Although individuality and individualism are commonly considered to mature with age/time and experience/wealth, a sane adult human being is usually considered by the state as an "individual person" in law, even if the person denies individual culpability ("I followed instru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial Of A Graph
A characteristic is a distinguishing feature of a person or thing. It may refer to: Computing * Characteristic (biased exponent), an ambiguous term formerly used by some authors to specify some type of exponent of a floating point number * Characteristic (significand), an ambiguous term formerly used by some authors to specify the significand of a floating point number Science *''I–V'' or current–voltage characteristic, the current in a circuit as a function of the applied voltage *Receiver operating characteristic Mathematics * Characteristic (algebra) of a ring, the smallest common cycle length of the ring's addition operation * Characteristic (logarithm), integer part of a common logarithm * Characteristic function, usually the indicator function of a subset, though the term has other meanings in specific domains * Characteristic polynomial, a polynomial associated with a square matrix in linear algebra * Characteristic subgroup, a subgroup that is invariant under all aut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1, or e depending on the context. If the group operation is denoted \, \cdot \, then it is defined by e \cdot e = e. The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. Definitions Given any group G, the group consisting of only the identity element is a subgroup of G, and, being the trivial group, is called the of G. The term, when referred to "G has no nontrivial proper subgroups" refers to the only subgroups of G being the trivial group \ and the group G itself. Properties The trivial group is cycl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The highest-ranking editor of a publication may also be titled editor, managing ... of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frucht's Theorem
Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group ''G'' there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to ''G''. Proof idea The main idea of the proof is to observe that the Cayley graph of ''G'', with the addition of colors and orientations on its edges to distinguish the generators of ''G'' from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have ''G'' as its symmetry group., discussion following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymmetric Graph
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation of its vertices with the property that any two vertices and are adjacent if and only if and are adjacent. The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms. Examples The smallest asymmetric non- trivial graphs have 6 vertices. The smallest asymmetric regular graphs have ten vertices; there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular. One of the five smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939.. According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs. Properties The class of asymmetric graphs is closed under complements: a graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LCF Notation
In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for each vertex, and the LCF notation specifies how far along the cycle each vertex's third neighbor is. A single graph may have multiple different representations in LCF notation. Description In a Hamiltonian graph, the vertices can be arranged in a cycle, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence of these numbers of positions, starting from an arbitrarily chosen vertex and written in square brackets. The numbers between the brackets are interpreted modulo ''N'', where ''N'' is the number of vertic ... [...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] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a - cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
