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 in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. In one restricted but very common sense of the term, a graph is an ordered pair comprising: a set of vertices (also called nodes or points); a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity, this type of object may be called precisely an undirected simple graph. In the edge , the vertices and are called the endpoints of the edge. The edge is said to join and and to be incident on and on . A vertex may exist in a graph and not belong to an edge. Multiple edges, not allowed under the definition above, are two or more edges that join the same two vertices. In one more general sense of the term allowing multiple edges, a graph is an ordered triple comprising: a set of vertices (also called nodes or points); a set of edges (also called links or lines); an incidence function mapping every edge to an unordered pair of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity, this type of object may be called precisely an undirected multigraph. A loop is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) which is not in . So to allow loops the definitions must be expanded.