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Graceful Labeling
In graph theory, a graceful labeling of a graph with edges is a labeling of its vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and inclusive. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001PostScript/ref> A graph which admits a graceful labeling is called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major conjecture in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC. It hypothesizes that all trees are graceful. It is still an open conjecture, although a related but weaker conjecture known as "Ringel's conjecture" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cactus Graph
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or (for nontrivial cactus) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle. Properties Cacti are outerplanar graphs. Every pseudotree is a cactus. A nontrivial graph is a cactus if and only if every block is either a simple cycle or a single edge. The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing an edge from the complete graph ''K''4. Triangular cactus A triangular cactus is a special type of cactus graph such that each cycle has length three and each edge belongs to a cycle. For instance, the friendship graphs, graphs formed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graceful Labeling
In graph theory, a graceful labeling of a graph with edges is a labeling of its vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and inclusive. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001PostScript/ref> A graph which admits a graceful labeling is called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major conjecture in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC. It hypothesizes that all trees are graceful. It is still an open conjecture, although a related but weaker conjecture known as "Ringel's conjecture" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caterpillar Graph
In graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by Arthur Hobbs. As colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed.". Equivalent characterizations The following characterizations all describe the caterpillar trees: *They are the trees for which removing the leaves and incident edges produces a path graph. *They are the trees in which there exists a path that contains every vertex of degree two or more. *They are the trees in which every vertex of degree at least three has at most two non-leaf neighbors. *They are the trees that do not contain as a subgraph the graph formed by replacing every edge in the star graph ''K''1,3 by a path of length two. *They are the connected graphs that can be drawn with their vertices on two parallel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called . The number of vertices in equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it. Terminology There are many synonyms for "cycle graph". These include simple cycle graph and cyclic graph, although the latter term is less often used, because it can also refer to graphs which are merely not acyclic. Among graph theorists, cycle, polygon, or ''n''-gon are also often used. The term ''n''-cycle is sometimes used in other settings. A cycle with an even number of vertices is called an even cycle; a cycle with an odd number of vertices is called an odd cycle. Properties A cycle graph is: * 2-edge colorable, if and only if it has an even number of vertices * 2-regular * 2- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 is called ''the'' unit hypercube. Construction A hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. Square grid graph A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gear Graph
This partial list of graphs contains definitions of graphs and graph families which are known by particular names, but do not have a Wikipedia article of their own. For collected definitions of graph theory terms that do not refer to individual graph types, such as ''vertex'' and ''path'', see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Graphs. Gear A gear graph, denoted ''G''''n'' is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph ''W''''n''. Thus, ''G''''n'' has 2''n''+1 vertices and 3''n'' edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels. Helm A helm graph, denoted ''Hn'' is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph ''Wn''. Lobster A lobster graph is a tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helm Graph
This partial list of graphs contains definitions of graphs and graph families which are known by particular names, but do not have a Wikipedia article of their own. For collected definitions of graph theory terms that do not refer to individual graph types, such as ''vertex'' and ''path'', see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Graphs. Gear A gear graph, denoted ''G''''n'' is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph ''W''''n''. Thus, ''G''''n'' has 2''n''+1 vertices and 3''n'' edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels. Helm A helm graph, denoted ''Hn'' is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph ''Wn''. Lobster A lobster graph is a tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheel Graph
A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be moved easily facilitating movement or transportation while supporting a load, or performing labor in machines. Wheels are also used for other purposes, such as a ship's wheel, steering wheel, potter's wheel, and flywheel. Common examples are found in transport applications. A wheel reduces friction by facilitating motion by rolling together with the use of axles. In order for wheels to rotate, a moment needs to be applied to the wheel about its axis, either by way of gravity or by the application of another external force or torque. Using the wheel, Sumerians invented a device that spins clay as a potter shapes it into the desired object. Terminology The English word ''wheel'' comes from the Old English word , from Proto-Germanic , from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
