|
Bivariegated Graph
In graph theory, a bivariegated graph is a graph whose vertex set can be partitioned into two equal parts such that each vertex is adjacent to exactly one vertex from the other set not containing it.... In a bivarigated graph ''G'' with 2''n'' vertices, there exists a set of ''n'' independent edges such that no odd number of them lie on a cycle of ''G''. Examples The Petersen graph, shown below, is a bivariegated graph: if one partitions it into an outer pentagon and an inner five-point star, each vertex on one side of the partition has exactly one neighbor on the other side of the partition. More generally, the same is true for any generalized Petersen graph formed by connecting an outer polygon and an inner star with the same number of points; for instance, this applies to the Möbius–Kantor graph and the Desargues graph. Any hypercube graph, such as the four-dimensional hypercube shown below, is also bivariegated. However, the graph shown below is not bivariegated. Whate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 connected by ''edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Graph
In the mathematics, mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertex (graph theory), vertices and 15 edge (graph theory), 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 Bridge (graph theory), 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 appearanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Petersen Graph
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins. Definition and notation In Watkins' notation, ''G''(''n'', ''k'') is a graph with vertex set :\ and edge set :\ where subscripts are to be read modulo ''n'' and ''k'' < ''n''/2. Some authors use the notation ''GPG''(''n'', ''k''). Coxeter's notation for the same graph would be + , a combination of the Schläfli symbols for the regular ''n''-gon and star polygon from which the graph is formed. The Petersen graph itself is ''G''(5, 2) or + . Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius–Kantor Graph
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph ''G''(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it. Möbius–Kantor configuration asked whether there exists a pair of polygons with ''p'' sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a projective configuration. For ''p'' = 4 there is no solution in the Euclidean plane, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane. That is, in Kantor's s ... [...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 additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube Graph
In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, edges, and is a regular graph with edges touching each vertex. The hypercube graph may also be constructed by creating a vertex for each subset of an -element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each -digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the -fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of connected to each other by a perfect matching. Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph that is a cubic graph is the cubical graph . Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Number
Independence is a condition of a person, nation, country, or Sovereign state, state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of a dependent territory. The commemoration of the independence day of a country or nation celebrates when a country is free from all forms of foreign colonialism; free to build a country or nation without any interference from other nations. Definition of independence Whether the attainment of independence is different from revolution has long been contested, and has often been debated over the question of violence as Legitimacy (family law), legitimate means to achieving sovereignty. In general, revolutions aim only to redistribute power with or without an element of emancipation,such as in democratization ''within'' a state, which as such may remain unaltered. For example, the Mexican Revolution (1910) chiefly refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactly one edge in . A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs: : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cover. If there is a perfect matching, then both the matching number and the edge cover number equal . A perfect matching can only occur when the graph has an even n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Petersen Graph")
%2C_by_John_Trumbull.jpg "Click for more on -> Independence Number")