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Johnson Graph
Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J(n,k) are the k-element subsets of an n-element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains (k-1)-elements.. Both Johnson graphs and the closely related Johnson scheme are named after Selmer M. Johnson. Special cases *J(n,1) is the complete graph . *J(4,2) is the octahedral graph. *J(5,2) is the complement graph of the Petersen graph, hence the line graph of . More generally, for all n, the Johnson graph J(n,2) is the complement of the Kneser graph K(n,2). Graph-theoretic properties * J(n,k) is isomorphic to J(n,n-k). * For all 0 \leq j \leq \operatorname(J(n,k)), any pair of vertices at distance j share k-j elements in common. * J(n,k) is Hamilton-connected, meaning that every pair of vertices forms the endpoints of a Hamiltonian path in the graph. In particular this means that it has a Hamiltonian cycle. * It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Graph J(5,2)
Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a habitational name. Etymology The name itself is a patronym of the given name ''John (first name), John'', literally meaning "son of John". The name ''John'' derives from Latin ''Johannes'', which is derived through Greek language, Greek ''Iōannēs'' from Hebrew ''Yohanan'', meaning "Yahweh has favoured". Origin The name has been extremely popular in Europe since the Christian era as a result of it being given to St John the Baptist, St John the Evangelist and nearly one thousand other Christian saints. Other Germanic languages * Swedish language, Swedish: Johnsson, Jonsson * Icelandic language, Icelandic: Jónsson See also * List of people with surname Johnson *Gjoni (Gjonaj) *Ioannou *Jensen (surname), Jensen *Johansson *Johns ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Path
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] |
Grassmann Graph
In graph theory, Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph are the -dimensional subspaces of an -dimensional vector space over a finite field of order ; two vertices are adjacent when their intersection is -dimensional. Many of the parameters of Grassmann graphs are -analogs of the parameters of Johnson graphs, and Grassmann graphs have several of the same graph properties as Johnson graphs. Graph-theoretic properties * is isomorphic to . * For all , the intersection of any pair of vertices at distance is -dimensional. * The clique number of is given by an expression in terms its least and greatest eigenvalues and : ::\omega \left ( J_q(n,k) \right ) = 1 - \frac Automorphism group There is a distance-transitive subgroup of \operatorname(J_q(n, k)) isomorphic to the projective linear group \operatorname(n, q). In fact, unless n = 2k or k \in \, \operatorname(J_q(n,k)) \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Graph
In the mathematical field of graph theory, the odd graphs ''On'' are a family of symmetric graphs with high odd girth, defined from certain set systems. They include and generalize the Petersen graph. Definition and examples The odd graph ''On'' has one vertex for each of the (''n'' − 1)-element subsets of a (2''n'' − 1)-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint. That is, ''On'' is the Kneser graph ''KG''(2''n'' − 1,''n'' − 1). ''O''2 is a triangle, while ''O''3 is the familiar Petersen graph. The generalized odd graphs are defined as distance-regular graphs with diameter ''n'' − 1 and odd girth 2''n'' − 1 for some ''n''. They include the odd graphs and the folded cube graphs. History and applications Although the Petersen graph has been known since 1898, its definition as an odd graph dates to the work of , who also stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \leq i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by A \ominus B, or A\operatorname \triangle B. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Properties The symmetric difference is equivalent to the union of both relative complements, that is: :A\,\triangle\,B = \left(A \setminus B\right) \cup \left(B \setminus A\right), The symmetric difference can also be expressed using the XOR operation ⊕ on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Definition An ''n''-class association scheme consists of a set ''X'' together with a partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \ and is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S'' *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \in R_ is a constant p^k_ depending on i, j, k but not on the particular choice of x and y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-regular Graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. Intersection arrays It turns out that a graph G of diameter d is distance-regular if and only if there is an array of integers \ such that for all 1 \leq j \leq d , b_j gives the number of neighbours of u at distance j+1 from v and c_j gives the number of neighbours of u at distance j - 1 from v for any pair of vertices u and v at distance j on G . The array of integers characterizing a distance-regular graph is known as its intersection array. Cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-transitive Graph
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices and at any distance , and any other two vertices and at the same distance, there is an automorphism of the graph that carries to and to . Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith. A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. Examples Some first examples of families of distance-transitive graphs include: * The Johnson graphs. * The Grassmann graphs. * The Hamming Graphs. * The folded cube graphs. * The square rook's graphs. * The hypercube graphs. * The Livingstone graph. Classification of cubic distance-transitive graphs After introducing them in 1971, Biggs and Smith showed that there are only 12 finite trivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Number
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Number
In the mathematical area of 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 bioinf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersimplex
In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones and d-k zeros. Equivalently, \Delta_ can be obtained by slicing the d-dimensional unit hypercube ,1d with the hyperplane of equation x_1+\cdots+x_d=k and, for this reason, it is a (d-1)-dimensional polytope when 0 Properties The number of vertices of is . The graph formed by the vertices and edges of the hypersimplex is the .Alternative constructions An alternative construction (for ) is to take the convex hull ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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