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Errera Graph
In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof of the four color theorem; it was named after Errera by . Properties The Errera graph is planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth 3. All its vertices are of degree 5 or 6 and it is a 5- vertex-connected graph and a 5- edge-connected graph. The Errera graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 20, the group of symmetries of a decagon, including both rotations and reflections. The characteristic polynomial of the Errera graph is -(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2. Application to the four color theorem The four color theorem states that the vertices of every planar graph can be colored with four colors, so that no two adjacent vertices have equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errera Graph Alt
Errera may refer to: People * Alberto Errera (1913–1944), Greek-Jewish officer and member of the anti-Nazi resistance * Alfred Errera (1886–1960), Belgian mathematician * Emilia Errera (1866–1901), Italian teacher and writer * Gérard Errera (born 1943), French diplomat * Isabelle Errera, née Goldschmidt (1869–1929), Belgian art historian specializing in textiles * Jacques Errera (1896–1977), Belgian physicochemist * Léo Errera (1858–1905), Belgian botanist * Nicolas Errèra (1967), French musician and composer * Rosa Errera (1864–1946), Italian writer, translator, and teacher Toponyms * Cape Errera, cape which forms the southwest end of Wiencke Island, in the Palmer Archipelago * Errera Channel, channel between Rongé Island and the west coast of Graham Land Other * Errera graph In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a in ''G''. The edge connectivity version of provi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster (physics)
In physics, the term clusters denotes small, polyatomic particles. As a rule of thumb, any particle made of between 3×100 and 3×107 atoms is considered a cluster. The term can also refer to the organization of protons and neutrons within an atomic nucleus, e.g. the alpha particle (also known as "α-cluster"), consisting of two protons and two neutrons (as in a helium nucleus). Overview Although first reports of cluster species date back to the 1940s, cluster science emerged as a separate direction of research in the 1980s, One purpose of the research was to study the gradual development of collective phenomena which characterize a bulk solid. For example, these are the color of a body, its electrical conductivity, its ability to absorb or reflect light, and magnetic phenomena such as ferro-, ferri-, or antiferromagnetism. These are typical collective phenomena which only develop in an aggregate of a large number of atoms. It was found that collective phenomena break down for v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gold
Gold is a chemical element with the symbol Au (from la, aurum) and atomic number 79. This makes it one of the higher atomic number elements that occur naturally. It is a bright, slightly orange-yellow, dense, soft, malleable, and ductile metal in a pure form. Chemically, gold is a transition metal and a group 11 element. It is one of the least reactive chemical elements and is solid under standard conditions. Gold often occurs in free elemental ( native state), as nuggets or grains, in rocks, veins, and alluvial deposits. It occurs in a solid solution series with the native element silver (as electrum), naturally alloyed with other metals like copper and palladium, and mineral inclusions such as within pyrite. Less commonly, it occurs in minerals as gold compounds, often with tellurium (gold tellurides). Gold is resistant to most acids, though it does dissolve in aqua regia (a mixture of nitric acid and hydrochloric acid), forming a soluble tetrachloroaurate anion. Gold is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typically ductile (can be drawn into wires) and malleable (they can be hammered into thin sheets). These properties are the result of the ''metallic bond'' between the atoms or molecules of the metal. A metal may be a chemical element such as iron; an alloy such as stainless steel; or a molecular compound such as polymeric sulfur nitride. In physics, a metal is generally regarded as any substance capable of conducting electricity at a temperature of absolute zero. Many elements and compounds that are not normally classified as metals become metallic under high pressures. For example, the nonmetal iodine gradually becomes a metal at a pressure of between 40 and 170 thousand times atmospheric pressure. Equally, some materials regarded as metals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge of has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of . The definition of the dual depends on the choice of embedding of the graph , so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and biochemistry, the distinction from ions is dropped and ''molecule'' is often used when referring to polyatomic ions. A molecule may be homonuclear, that is, it consists of atoms of one chemical element, e.g. two atoms in the oxygen molecule (O2); or it may be heteronuclear, a chemical compound composed of more than one element, e.g. water (two hydrogen atoms and one oxygen atom; H2O). In the kinetic theory of gases, the term ''molecule'' is often used for any gaseous particle regardless of its composition. This relaxes the requirement that a molecule contains two or more atoms, since the noble gases are individual atoms. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are typically not consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Graph Theory
Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo Hosoya, Milan Randić and Nenad Trinajstić (also Harry Wiener and others). In 1988, it was reported that several hundred researchers worked in this area, producing about 500 articles annually. A number of monographs have been written in the area, including the two-volume comprehensive text by Trinajstić, ''Chemical Graph Theory'', that summarized the field up to mid-1980s. The adherents of the theory maintain that the properties of a chemical graph (i.e., a graph-theoretical representation of a molecule) give valuable insights into the chemical phenomena. Others contend that graphs play only a fringe role in chemical research.D.H. Rouvray, "Combinatorics in Chemistry", pp. 1955-1982, in: Ronald Graham, Martin Grötschel, László Lovász (E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kempe Chain
Kempe may refer to: * Kempe baronets, a title in the Baronetage of England * Kempe chain, part of the four-colour theorem * Kempe Fjord, King Christian X Land, Greenland * Kempe Glacier, Antarctica * Kempe Hill, former name of Camp Hill, West Midlands, England People with the surname * Alfred Kempe (1849–1922), English mathematician * Arnold E. Kempe (born 1927), American lawyer and politician * Carl Kempe (1884–1967), Swedish paper producer * Charles Eamer Kempe (1837–1907), English stained glass designer * C. Henry Kempe (1922–1984), American pediatrician who identified the Battered child syndrome * Kempe Gowda I (1513–69), Yelahanka chieftain, founded the city of Bangalore * Margery Kempe (c. 1373–after 1438), English autobiographer, religious pilgrim * Raymond J. Kempe (born 1931), American lawyer and politician * Rudolf Kempe (1910–76), German conductor * William Kempe (died c. 1603), English actor and morris dancer See also * Kemp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poussin Graph
In graph theory, the Poussin graph is a planar graph with 15 vertices and 39 edges. It is named after Charles Jean de la Vallée-Poussin. History In 1879, Alfred Kempe published a proof of the four color theorem, one of the big conjectures in graph theory. While the theorem is true, Kempe's proof is incorrect. Percy John Heawood illustrated it in 1890 with a counter-example, and de la Vallée-Poussin reached the same conclusion in 1896 with the Poussin graph. Kempe's (incorrect) proof is based on alternating chains, and as those chains prove useful in graph theory mathematicians remain interested in such counterexamples. More were found later: first, the Errera graph in 1921, then the Kittell graph in 1935, with 23 vertices, and finally two minimal counter-examples (the Soifer graph in 1997 and the Fritsch graph in 1998, both of order 9).Gethner, E. and Springer, W. M. II. « How False Is Kempe's Proof of the Four-Color Theorem? » Congr. Numer. 164, 159–175, 2003. Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Errera Kempe Chains
Errera may refer to: People * Alberto Errera (1913–1944), Greek-Jewish officer and member of the anti-Nazi resistance * Alfred Errera (1886–1960), Belgian mathematician * Emilia Errera (1866–1901), Italian teacher and writer * Gérard Errera (born 1943), French diplomat * Isabelle Errera, née Goldschmidt (1869–1929), Belgian art historian specializing in textiles * Jacques Errera (1896–1977), Belgian physicochemist * Léo Errera (1858–1905), Belgian botanist * Nicolas Errèra (1967), French musician and composer * Rosa Errera (1864–1946), Italian writer, translator, and teacher Toponyms * Cape Errera, cape which forms the southwest end of Wiencke Island, in the Palmer Archipelago * Errera Channel, channel between Rongé Island and the west coast of Graham Land Other * Errera graph In the mathematical field of graph theory, the Errera graph is a graph with 17 vertices and 45 edges. Alfred Errera published it in 1921 as a counterexample to Kempe's erroneous proof o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
