|
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic, often denoted by G\simeq H. In the case when the isomorphism is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the isomorphism is called an automorphism of ''G''. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitneys Theorem Exception
The Whitney Awards are awards given annually for novels by LDS authors. Established in 2007, they are named after Orson F. Whitney, a prominent early member of the LDS Church. There are several categories for which novels may be nominated. The Whitney Awards are a semi-independent non-profit organization affiliated with the LDStorymakers, a guild for LDS authors. Categories Due to the limited number of titles released by LDS authors, several of the genre awards have been combined (such as romance and women's fiction). , there are eight genre categories: Adult * Romance * Mystery/Suspense * Speculative *Historical *General Youth * Youth General *Youth Speculative * Middle Grade There are also two special awards: *Best Novel by a New Author *Novel of the Year The Whitney Committee states that it is unlikely that other areas of LDS art—such as music, poetry, or non-fiction books—will be added to the categories. Process To be eligible, a novel must be written by an L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. * ''n'' is called the length of the circuit resp. length of the cycle. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labeling is a function of to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labeling is a function of to a set of labels. In this case, the graph is called an edge-labeled graph. When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph. When used without qualification, the term labeled graph generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers , where is the number of vertices in the graph. For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned weights representing the "cost" of trave ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (data Structure)
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data structure consists of a finite (and possibly mutable) set of ''vertices'' (also called ''nodes'' or ''points''), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as ''edges'' (also called ''links'' or ''lines''), and for a directed graph are also known as ''edges'' but also sometimes ''arrows'' or ''arcs''. The vertices may be part of the graph structure, or may be external entities represented by integer indices or references. A graph data structure may also associate to each edge some ''edge value'', such as a symbolic label or a numeric attribute (cost, capacity, length, etc.). Operations The basic operations provided by a graph data structure ''G'' usually include:See, e.g. , Sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Properties
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.. Definitions While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible isomorphisms of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term "graph invariant" is used for properties expressed quantitatively, while "property" usually refers to descriptive characterizations of graphs. For example, the statement "graph does not have vertices of degree 1" is a "property" while "the number of vertices of degree 1 in a graph" is an "invariant". More formally, a graph property is a class of graphs with the property that any two isomorphic gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rooted Tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colored Graph
''Colored'' (or ''coloured'') is a racial descriptor historically used in the United States during the Jim Crow era to refer to an African Americans, African American. In many places, it may be considered a Pejorative, slur. Dictionary definitions The word ''colored'' was first used in the 14th century but with a meaning other than race or ethnicity. The earliest uses of the term to denote a member of dark-skinned groups of peoples occurred in the second part of the 18th century in reference to South America. According to the ''Oxford English Dictionary'', "colored" was first used in this context in 1758 to translate the Spanish term () in Antonio de Ulloa's ''A voyage to South America''. The term came in use in the United States during the early 19th century, and it then was adopted by emancipated slaves as a term of racial pride after the end of the American Civil War until it was replaced as a self-designation by ''Black'' or ''African-American'' during the second part of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
