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Zero-symmetric Graph
In the mathematics, mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique graph automorphism, symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge. The name for this class of graphs was coined by R. M. Foster in a 1966 letter to Harold Scott MacDonald Coxeter, H. S. M. Coxeter. In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.. Examples The smallest zero-symmetric graph is a nonplanar graph with 18 vertices. Its LCF notation is [5,−5]9. Among planar graphs, the truncated cuboctahedral graph, truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric. These examples are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Cuboctahedron
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: \Bigl(\pm 1, \quad \pm\left(1 + \sqrt 2\right), \quad \pm\left(1 + 2\sqrt 2\right) \Bigr). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are: :\begin A &= 12\left(2+\sqrt+\sqrt\right) a^2 &&\approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Cuboctahedral Graph
In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a Truncation (geometry), truncation of a cuboctahedron. It has 12 Square (geometry), square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can Omnitruncated cubic honeycomb, tessellate with the octagonal prism. Names There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron. Cartesian coordinates The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of: \Bigl(\pm 1, \quad \pm\left(1 + \sqrt 2\right), \quad \pm\left(1 + 2\sqrt 2\right) \Bigr). Area and volume The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge leng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ''D'' wil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Semi-symmetric Graph
In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of the graph's edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the first into the second. Properties A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition (in fact, regularity is not required for this property to hold). For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other. History Semi-symmetric graphs were first studied E. Dauber, a student of F. Harary, in a paper, no longer available, titled "On line- but not point-sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lovász Conjecture
In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: : Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposite way, but this version became standard. In 1996, László Babai published a conjecture sharply contradicting this conjecture, but both conjectures remain widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples. Historical remarks The problem of finding Hamiltonian paths in highly symmetric graphs is quite old. As Donald Knuth describes it in volume 4 of ''The Art of Computer Programming'', the problem originated in British campanology (bell-ringing). Such Hamiltonian paths and cycles are also closely connected to Gray codes. In each case the constructions are explicit. Variants of the Lovász conjecture Hamiltonian cycle Another version of Lovász conjecture states tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hamiltonian Cycle
In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle (graph theory), 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. The computational problems of determining whether such paths and cycles exist in graphs are NP-complete; see Hamiltonian path problem for details. 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 structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Combinatorial Enumeration
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are '' closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified generating set of a group, set of generators for the group. It is a central tool in combinatorial group theory, combinatorial and geometric group theory. The structure and symmetry of Cayley graphs make them particularly good candidates for constructing expander graphs. Definition Let G be a group (mathematics), group and S be a generating set of a group, generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an Edge coloring, edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truncated Icosidodecahedral Graph
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,Wenninger Model Number 16 great rhombicosidodecahedron,Williams (Section 3-9, p. 94)Cromwell (p. 82) omnitruncated dodecahedron or omnitruncated icosahedronNorman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966 is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces. It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
