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Locally Linear Graph
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighborhood (graph theory), neighbors are each adjacent to exactly one other neighbor. That is, locally (from the point of view of any one vertex) the rest of the graph looks like a perfect matching. Locally linear graphs have also been called locally matched graphs. More technically, the triangles of any locally linear graph form the hyperedges of a triangle-free 3-uniform linear hypergraph, and they form the blocks of certain Steiner system, partial Steiner triple systems; and the locally linear graphs are exactly the Gaifman graphs of these hypergraphs or partial Steiner systems. Many constructions for locally linear graphs are known. Examples of locally linear graphs include the triangular cactus graphs, the line graphs of 3-regular triangle-free graphs, and the Cartesian product of graphs, Cartesian products of sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cage (graph Theory)
In the mathematics, mathematical field of graph theory, a cage is a regular graph that has as few vertex (graph theory), vertices as possible for its girth (graph theory), girth. Formally, an is defined to be a graph (discrete mathematics), graph in which each vertex has exactly neighbors, and in which the shortest cycle (graph theory), cycle has a length of exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with parity (mathematics), odd girth must have at least :1 + r\sum_^(r-1)^i vertices, and any cage with parity (mathematics), even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three- edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Graph
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a normative context, utility refers to a goal or objective that we wish to maximize, i.e., an objective function. This kind of utility bears a closer resemblance to the original utilitarian concept, developed by moral philosophers such as Jeremy Bentham and John Stuart Mill. * In a descriptive context, the term refers to an ''apparent'' objective function; such a function is revealed by a person's behavior, and specifically by their preferences over lotteries, which can be any quantified choice. The relationship between these two kinds of utility functions has been a source of controversy among both economists and ethicists, with most maintaining that the two are distinct but generally related. Utility function Consider a set of alternatives among which a person has a preference ordering. A utility functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is Cuboctahedron#Radial equilateral symmetry, radially equilateral. Its dual polyhedron is the rhombic dodecahedron. Construction The cuboctahedron can be constructed in many ways: * Its construction can be started by attaching two regular triangular cupolas base-to-base. This is similar to one of the Johnson solids, triangular orthobicupola. The difference is that the triangular orthobicupola is constructed with one of the cupolas twisted so that similar polygonal faces are adjacent, whereas the cuboctahedron is not. As a result, the cuboctahedron may also called the ''triangular gyro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Hamming Graph
Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let be a set of elements and a positive integer. The Hamming graph has vertex set , the set of ordered -tuples of elements of , or sequences of length from . Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph is, equivalently, the Cartesian product of complete graphs .. In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.. Unlike the Hamming graphs , the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive. Special cases *, which is the generalized quadrangle *, which is the complete graph . *, which is the lattice graph and also the rook's graph *, which is the singleton gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-3 Duoprism
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duoprism is the simplest among them, and it can be constructed using Cartesian product of two triangles. The resulting duoprism has 9 vertices, 18 edges, and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram , and symmetry , order 72. The hypervolume of a uniform 3-3 duoprism with edge length a is V_4 = a^4. This is the square of the area of an equilateral triangle, A = a^2. The 3-3 duoprism can be represented as a graph with the same number of vertices and edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley Graph
In mathematics, Paley graphs are undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrix, conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrix, Hadamard matrices from quadratic residues. They were introduced as graphs independently by and . Horst Sachs, Sachs was interested in them for their self-complementarity properties, while Paul Erdős, Erdős and Alfréd Rényi, Rényi studied their symmetries. Paley digraphs are directed graph, directed analogs of Paley graphs that yield antisymmetric conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-sum
In graph theory, a branch of mathematics, a clique sum (or clique-sum) is a way of combining two graphs by gluing them together at a clique (graph theory), clique, analogous to the connected sum operation in topology. If two graphs ''G'' and ''H'' each contain cliques of equal size, the clique-sum of ''G'' and ''H'' is formed from their disjoint union of graphs, disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then deleting all the clique edges (the original definition, based on the notion of set sum) or possibly deleting some of the clique edges (a loosening of the definition). A ''k''-clique-sum is a clique-sum in which both cliques have exactly (or sometimes, at most) ''k'' vertices. One may also form clique-sums and ''k''-clique-sums of more than two graphs, by repeated application of the clique-sum operation. Different sources disagree on which edges should be removed as part of a clique-sum operation. In some contexts, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
