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Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of A Function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a Plane curve, curve. The graphical representation of the graph of a Function (mathematics), function is also known as a ''Plot (graphics), plot''. In the case of Bivariate function, functions of two variables – that is, functions whose Domain of a function, domain consists of pairs (x, y) –, the graph usually refers to the set of ordered triples (x, y, z) where f(x,y) = z. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a Surface (mathematics), surface, which can be visualized as a ''surface plot (graphics), surface plot''. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Water, Gas, And Electricity
The three utilities problem, also known as water, gas and electricity, is a mathematical puzzle that asks for non-crossing connections to be drawn between three houses and three utility companies on a plane. When posing it in the early 20th century, Henry Dudeney wrote that it was already an old problem. It is an impossible puzzle: it is not possible to connect all nine lines without any of them crossing. Versions of the problem on nonplanar surfaces such as a torus or Möbius strip, or that allow connections to pass through other houses or utilities, can be solved. This puzzle can be formalized as a problem in topological graph theory by asking whether the complete bipartite graph K_, with vertices representing the houses and utilities and edges representing their connections, has a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that K_ is not a planar graph. Multiple proofs of this impossibility are known, and form part of the proof of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Graph
In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive (see below). The Gray graph has chromatic number 2, chromatic index 3, radius 6 and diameter 6. It is also a 3- vertex-connected and 3- edge-connected non-planar graph. Construction The Gray graph can be constructed from the 27 points of a 3 × 3 × 3 grid and the 27 axis-parallel lines through these points. This collection of points and lines forms a projective configuration: each point has exactly three lines through it, and each line has exactly three points on it. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Biggs–Smith Graph
In the mathematical field of graph theory, the Biggs–Smith graph is a 3-regular graph with 102 vertices and 153 edges. It has chromatic number 3, chromatic index 3, radius 7, diameter 7 and girth 9. It is also a 3- vertex-connected graph and a 3- edge-connected graph. All the cubic distance-regular graphs are known. The Biggs–Smith graph is one of the 13 such graphs. Algebraic properties The automorphism group of the Biggs–Smith graph is a group of order 2448 isomorphic to the projective special linear group PSL(2,17). It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Biggs–Smith graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the ''Foster census'', the Biggs–Smith graph, referenced as F102A, is the only cubic symmetric graph on 102 vertices. The Biggs–Smith graph is also uniquely determined by its graph spectrum, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foster Graph
In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has chromatic number 2, chromatic index 3, radius 8, diameter 8 and girth 10. It is also a 3- vertex-connected and 3- edge-connected graph. It has queue number 2 and the upper bound on the book thickness is 4. All the cubic distance-regular graphs are known. The Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array . It can be constructed as the incidence graph of the partial linear space which is the unique triple cover with no 8-gons of the generalized quadrangle ''GQ''(2,2). It is named after R. M. Foster, whose ''Foster census'' of cubic symmetric graphs included this graph. The bipartite half of the Foster graph is a distance-regular graph and a locally linear graph. It is one of a finite number of such graphs with degree six. Algebraic properties The automo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyck Graph
In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian with 120 distinct Hamiltonian cycles. It has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3- vertex-connected and a 3- edge-connected graph. It has book thickness 3 and queue number 2. Algebraic properties The automorphism group of the Dyck graph is a group of order 192. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Dyck graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the ''Foster census'', the Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices. The characteristic polynomial of the Dyck graph is equal to (x-3) (x-1)^9 (x+1)^9 (x+3) (x^2-5)^6. Toroidal graph The Dyck graph is a toroidal graph, contained in the skeleto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte–Coxeter Graph
In the mathematics, mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth (graph theory), girth 8, it is a cage (graph theory), cage and a Moore graph. It is bipartite graph, bipartite, and can be constructed as the Levi graph of the generalized quadrangle ''W''2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a). All the cubic graph, cubic distance-regular graphs are known. The Tutte–Coxeter is one of the 13 such graphs. It has Crossing number (graph theory), crossing number 13, book thickness 3 and queue number 2.Wolz, Jessica; ''Engineering Linear Layouts with SAT.'' Master Thesis, Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Graph
In the mathematics, mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic graph, cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter. Properties The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth (graph theory), girth 7. It is also a 3-k-vertex-connected graph, vertex-connected graph and a 3-k-edge-connected graph, edge-connected graph. It has book thickness 3 and queue number 2. The Coxeter graph is hypohamiltonian graph, hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has Crossing number (graph theory), rectilinear crossing number 11, and is the smallest cubic graph with that crossing number . Construction The simplest construction of a Coxeter graph is from a Fano plane. Take the Combination, 7C3 = 35 possible 3-combinations on 7 obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nauru Graph
In the mathematical field of graph theory, the Nauru graph is a symmetric, bipartite, cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru. It has chromatic number 2, chromatic index 3, diameter 4, radius 4 and girth 6. Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002. It is also a 3- vertex-connected and 3- edge-connected graph. It has book thickness 3 and queue number 2. The Nauru graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the McGee graph, also known as the (3-7)-cage. Construction The Nauru graph is Hamiltonian and can be described by the LCF notation : , −9, 7, −7, 9, −5sup>4. Eppstein, D.The many faces of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases. The name "Desargues graph" has also been used to refer to a ten-vertex graph, the complement of the Petersen graph, which can also be formed as the bipartite half of the 20-vertex Desargues graph. Constructions There are several different ways of constructing the Desargues graph: *It is the generalized Petersen graph . To form the Desargues graph in this way, connect ten of the vertices into a regular decagon, and connect the other ten vertices into a ten-pointed star that connects pairs of vertices at distance three in a second decagon. The Desargues graph consists of the 20 edges of these two polygons together with an additional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Graph
In the mathematical field of graph theory, the Pappus graph is a bipartite, 3- regular, undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic, distance-regular graphs are known; the Pappus graph is one of the 13 such graphs. The Pappus graph has rectilinear crossing number 5, and is the smallest cubic graph with that crossing number . It has girth 6, diameter 4, radius 4, chromatic number 2, chromatic index 3 and is both 3- vertex-connected and 3- edge-connected. It has book thickness 3 and queue number 2. The Pappus graph has a chromatic polynomial equal to: (x-1)x(x^ - 26x^ + 325x^ - 2600x^ + 14950x^ - 65762x^ + 229852x^ - 653966x^9 + 1537363x^8 - 3008720x^7 + 4904386x^6 - 6609926x^5 + 7238770x^4 - 6236975x^3 + 3989074x^2 - 1690406x + 356509) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
