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In mathematics and computer science, a graph is the basic object of study in graph theory. Informally speaking, a graph is a set of objects called points or vertices connected by links called lines or edges. In a graph proper, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A. In a digraph, short for directed graph, the two directions are counted as being distinct arcs or directed edges. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges). Definitions Definitions in graph theory vary in the literature. The following are some of the more basic ways of defining graphs and related structures. Graph A graph or undirected graph is an ordered pair that is subject to the following conditions:
(and hence ) are usually taken to be finite sets, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is (the number of vertices). A graph's size is , the number of edges. The degree of a vertex is the number of other vertices it is connected to by edges. Directed graph A directed graph or digraph is an ordered pair with An edge is considered to be directed from to ; is called the head and is called the tail of the edge; is said to be a direct successor of , and is said to be a direct predecessor of . If a path leads from to , then is said to be a successor of , and is said to be a predecessor of . A variation on this definition is the oriented graph, which is a graph (or multigraph; see below) with an orientation or direction assigned to each of its edges. A distinction between a directed graph and an oriented simple graph is that if and are vertices, a directed graph allows both and as edges, while only one is permitted in an oriented graph. A more fundamental difference is that, in a directed graph (or multigraph), the directions are fixed, but in an oriented graph (or multigraph), only the underlying graph is fixed, while the orientation may vary. A directed acyclic graph, also called a dag or DAG, is a directed graph with no directed cycles. A quiver is sometimes said to be simply a directed graph, but in practice it is a directed graph with vector spaces attached to the vertices and linear transformations attached to the arcs. Mixed graph A mixed graph is an ordered triple with , and defined as above. Variations in the definitions As defined above, edges of undirected graphs have two distinct ends, and E and A are sets (with distinct elements, like all sets). Many applications require more general possibilities, but terminology varies. A loop is an edge (directed or undirected) with both ends the same; these may be permitted or not permitted according to the application. In this context, an edge with two different ends is called a link. Sometimes and are allowed to be multisets, so that there can be more than one edge (called multiple edges) between the same two vertices. Another way to allow multiple edges is to make a set, independent of , and to specify the endpoints of an edge by an incidence relation between and . The same applies to a directed edge set , except that there must be two incidence relations, one for the head and one for the tail of each edge. A simple graph is always understood to mean a "graph" which contains no loops or multiple edges. However, the unqualified term "graph" might allow or disallow loops and multiple edges in the literature, according to the preferences of the author and the requirements of the particular topic. The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. Where graphs are defined so as to allow loops and multiple edges, a multigraph is often defined to mean a graph without loops, however, where graphs are defined so as to disallow loops and mutiple edges, the term is often defined to mean a "graph" which can have both multiple edges and loops, although many use the term "pseudograph" for this meaning. In exceptional situations it is even necessary to have edges with only one end, called halfedges, or no ends (loose edges); see for example signed graphs. Properties of graphs For more definitions see Glossary of graph theory. Two edges of a graph are called adjacent (sometimes coincident) if they share a common vertex. Similarly, two vertices are called adjacent if they share a common edge, in 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 with 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 graph, but not all mathematicians allow this object. In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal routing problems such as the traveling salesman problem. 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; then the graph may be called unlabeled. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). If vertices are indistinguishable they may be distinguished by giving each vertex a label, hence the name vertex-labeled graph. The same remarks apply to edges, so that graphs which have labeled edges are called edge-labeled graphs. 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 unlabelled. (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.) Examples The picture is a graphic representation of the following graph The fact that vertex 1 is adjacent to vertex 2 is sometimes denoted by 1 ~ 2. Important graphs Basic examples are: More advanced kinds of graphs are: Operations on graphs There are several operations that produce new graphs from old ones. Unary operations Binary operations Generalizations In a hypergraph, an edge can 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. Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs. In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number. Notes See also | |||||||
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