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Matchstick Graph
In geometric graph theory, a branch of mathematics, a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other. That is, it is a graph that has an embedding which is simultaneously a unit distance graph and a plane graph. For this reason, matchstick graphs have also been called planar unit-distance graphs. Informally, matchstick graphs can be made by placing noncrossing matchsticks on a flat surface, hence the name. Regular matchstick graphs Much of the research on matchstick graphs has concerned regular graphs, in which each vertex has the same number of neighbors. This number is called the degree of the graph. Regular matchstick graphs can have degree 0, 1, 2, 3, or 4. The complete graphs with one, two, and three vertices (a single vertex, a single edge, and a triangle) are all matchstick graphs and are 0-, 1-, and 2-regular respectively. The smallest 3-regular matchstick graph is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Matchstick Graph
Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system where the unit cell is in the shape of a cube * Cubic function, a polynomial function of degree three * Cubic equation, a polynomial equation (reducible to ''ax''3 + ''bx''2 + ''cx'' + ''d'' = 0) * Cubic form, a homogeneous polynomial of degree 3 * Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 * Cubic plane curve (mathematics), a plane algebraic curve ''C'' defined by a cubic equation * Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity * Cubic surface, an algebraic surface in three-dimensional space * Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra * CUBIC, a histology method Computing * Cubic IDE, a modular deve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heiko Harborth
Heiko Harborth (born 11 February 1938, in Celle, Germany)Harborth's web site http://www.mathematik.tu-bs.de/harborth/ . Accessed 14 May 2009. is Professor of Mathematics at Braunschweig University of Technology, 1975–present, and author of more than 188 mathematical publications.AMS MathSciNet http://www.ams.org/mathscinet . Accessed 14 May 2009. His work is mostly in the areas of number theory, combinatorics and discrete geometry, including graph theory. Career Harborth has been an instructor or professor at Braunschweig University of Technology since studying there and receiving his PhD in 1965 under Hans-Joachim Kanold. Harborth is a member of the New York Academy of Sciences, Braunschweigische Wissenschaftliche Gesellschaft, the Institute of Combinatorics and its Applications, and many other mathematical societies. Harborth currently sits on the editorial boards of Fibonacci Quarterly, Geombinatorics, Integers: Electronic Journal of Combinatorial Number Theory. He serv ... [...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] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's menta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Graph
In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). As Dynkin diagrams In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group of ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Graph
In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also known as the cycle graph C_3 and the complete graph K_3. Properties The triangle graph has chromatic number 3, chromatic index 3, radius 1, diameter 1 and girth 3. It is also a 2- vertex-connected graph and a 2- edge-connected graph. Its chromatic polynomial is (x-2)(x-1)x. See also * 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 ... References {{reflist Individual graphs Regular graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw (graph Theory)
In graph theory, a star is the complete bipartite graph a tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order with maximum diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
Hamiltonian Cycle
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Theory Of The Reals
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form \exists X_1 \cdots \exists X_n \, F(X_1,\dots, X_n), where the variables X_i are interpreted as having real number values, and where F(X_1,\dots X_n) is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F, make it become true.. The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such sentence, whether it is true or false. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE. Thus it has significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Graph Algorithms And Applications
The ''Journal of Graph Algorithms and Applications'' is an open access peer-reviewed scientific journal covering the subject of graph algorithms and graph drawing. The journal was established in 1997 and the editor-in-chief is Giuseppe Liotta (University of Perugia). It is abstracted and indexed by Scopus and MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links .... MathSciNet, retrieved 2011-03-02. References External links *{{Official website, http://jgaa.info/ ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
