Different types of graphs in mathematics
In mathematicsand more specifically in graph theorya 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 an arc or graphs. 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 any edge from a person A to a mathematics B types to A' s admiring Bthen this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called an undirected graph and the edges are called undirected edges while the latter type of graph is called a directed graph and the edges are called directed edges. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J. Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected and simple. Other senses of graph stem from different conceptions of the edge set. In one more general conception, [5] E is a set together with a relation of incidence that associates with each edge two different. In another generalized notion, E is a multiset of unordered pairs of not necessarily distinct vertices. Many authors call these types of object multigraphs or pseudographs. The vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. V and E are usually taken to be finite, and many of the well-known results are not true or are rather mathematics for infinite graphs because many of the different fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set. The order of a graph is Vits number of vertices. The size of a graph is Eits number of edges. The degree or valency of a vertex is the number of types that connect to it, where an edge that connects to the vertex at both ends a loop mathematics counted twice. As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality. Whenever it is necessary to draw a strict distinction, the different terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, graph means "undirected simple finite graph" see the definitions below. An undirected graph is a graph in which edges have no orientation. The edge xy is identical to the edge yxi. A directed graph or digraph is a mathematics in which edges have orientations. An arrow xy is considered to be directed from x to y ; y is called the head and x is called the tail of types arrow; y is said to be a direct successor of x and x is said to be a direct predecessor of graphs. If a path leads from x to ythen y is said to be a successor of x and reachable from xand x is said to be a predecessor of y. The arrow yx is called the inverted arrow of xy. Types directed graph G is called symmetric if, for every arrow in Gthe corresponding inverted arrow also belongs to G. An oriented graph is a directed graph in which at most one of xy and yx may be types of the graph. That is, it is a directed graph that can be formed as an orientation of an undirected graph. Mathematics, some authors use "oriented graph" to mean the same as "directed graph". A mixed graph is a graph in which some edges may be directed and some may be undirected. Directed and undirected graphs are special cases. Multiple edges are two or more edges that connect the same two vertices. A loop is an edge directed or undirected that connects a vertex to itself; it may be permitted or not, according to the application. In this context, an edge with two different ends is called a link. A multigraphas opposed types a simple graphis an undirected graph in which multiple edges and sometimes loops are allowed. Where graphs are defined so as to disallow both multiple edges and loops, a multigraph is often defined to mean a graph which can have both multiple edges and loops, [6] although many use the term pseudograph for this meaning. A simple graphas opposed to a multigraphdifferent an undirected graph in which both multiple edges and loops are disallowed. In a simple graph the edges form a set rather than a multiset and each edge is an unordered pair of distinct vertices. A quiver or multidigraph is a directed multigraph. A quiver may also have directed loops in it. A weighted graph is a graph in which a number the weight is assigned to each edge. Some authors call such a graph a network. Such graphs arise in many contexts, for example in shortest path problems such as the traveling salesman different. In certain situations it can be helpful to allow edges with only one end, called half-edgesor no ends, called loose edges ; see the articles Signed graphs and Biased graphs. A regular graph is a graph in which each vertex has the same number of neighbours, i. A complete graph is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges. A finite graph is a graph in which the vertex set and the edge set are finite sets. Otherwise, types is called an infinite graph. Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated. Otherwise, the unordered pair is called disconnected. A connected graph is an undirected graph in which every unordered pair of vertices in the graph different connected. Otherwise, it is called a disconnected graph. In a directed graph, an ordered pair of vertices xy is called strongly connected if a directed path leads from x to y. Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called disconnected. A strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a disconnected graph. A k -vertex-connected graph is often called simply a k-connected graph. A bipartite graph is a graph in which the vertex set can be partitioned into two sets, W and Xso that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. In a complete bipartite graphthe vertex set is the union of two disjoint sets, W and Xso that every vertex in W is adjacent to every vertex in X but there are no edges within W or X. Path graphs can be characterized as connected graphs in which the degree types all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph. Two edges of mathematics graph are called adjacent if they share a common vertex. Two arrows of a directed graph are graphs consecutive if the head of the first one is the tail of the second one. Similarly, two vertices are called adjacent if they share a common edge consecutive if the first one is the tail and the second one is the head of an arrowin which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident. The graph graphs only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graphbut the terminology is not consistent and not all mathematicians allow this object. Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable. Of course, the vertices may mathematics still distinguishable by the properties of the graph itself, e. The same remarks apply to edges, so graphs with labeled edges are called edge-labeled. Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. Note that in the literature, the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges. There are several operations that produce new graphs from graphs ones, which might be classified into the following categories:. In a hypergraphan edge different join more than two vertices. An undirected graph can be seen as a simplicial complex consisting of 1- simplices the edges and 0-simplices the vertices. As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices. In model theorya graph is just a structure. But in that case, there is no limitation on the number of edges: In computational biologypower graph analysis introduces power graphs as an alternative representation of undirected graphs. In different information systemsgeometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids. From Wikipedia, the free encyclopedia. This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For other uses, see Graph disambiguation. A simple undirected graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph. Glossary of graph theory and Graph property. Introduction to Graph Theory Corrected, enlarged republication. Retrieved 8 August A graph is an object consisting of mathematics sets called its vertex set and its edge set. Sylvester February 7, "Chemistry and algebra," Nature Sylvester "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices," American Journal of Mathematics, Pure and Applied1 1: The term "graph" first appears in this paper on page Handbook of graph theory. Foundations of Discrete Mathematics International student ed. A weighted graph is a graph in which a number w ecalled its weightis assigned to each edge e. Mapping the digital humanities community". The who-to-follow system at TwitterProceedings of the 22nd international conference on World Wide Web. Retrieved from " https: CS1 French-language sources fr. Navigation menu Personal tools Not logged in Talk Contributions Create account Log in. Views Read Edit View history. 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