In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by 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. 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 this graph is directed, because owing money is not necessarily reciprocated. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image).
Definitions
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph
A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs { v 1 , v 2 } {\displaystyle \{v_{1},v_{2}\}} of vertices, whose elements are called edges (sometimes links or lines). An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of a graph is its number |V| of vertices, usually denoted by n. The size of a graph is its number |E| of edges, typically denoted by m. However, in some contexts, such as for expressing the computational complexity of algorithms, the term size is used for the quantity |V| + |E| (otherwise, a non-empty graph could have size 0). The degree or valency of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice. The vertices u and v of an edge {u, v} are called the edge's endpoints. The edge is said to join u and v and to be incident on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called isolated. When an edge { u , v } {\displaystyle \{u,v\}} exists, the vertices u and v are called adjacent. A multigraph is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs. Sometimes, graphs are allowed to contain loops, which are edges that join a vertex to itself. To allow loops, the pairs of vertices in E must be allowed to have the same node twice. Such generalized graphs are called graphs with loops or simply graphs when it is clear from the context that loops are allowed. Generally, the vertex set V is taken to be finite (which implies that the edge set E is also finite). Sometimes infinite graphs are considered, but they are usually viewed as a special kind of binary relation, because most results on finite graphs either do not extend to the infinite case or need a rather different proof. In a graph of order n, the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops are allowed). The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Specifically, two vertices x and y are adjacent if {x, y} is an edge. A graph is fully determined by its adjacency matrix A, which is an n × n square matrix, with Aij specifying the number of connections from vertex i to vertex j. For a simple graph, Aij is either 0, indicating disconnection, or 1, indicating connection; moreover Aii = 0 because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all Aii being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all Aij being equal to a positive integer. Undirected graphs will have a symmetric adjacency matrix (meaning Aij = Aji).
Directed graph
A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, a directed graph is a pair G = (V, E) comprising:
V, a set of vertices (also called nodes or points); E, a set of edges (also called directed edges, directed links, directed lines, arrows, or arcs), which are ordered pairs of distinct vertices: E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 and x ≠ y } {\displaystyle E\subseteq \{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\}} . To avoid ambiguity, this type of object may be called precisely a directed simple graph. In the edge (x, y) directed from x to y, the vertices x and y are called the endpoints of the edge, x the tail of the edge and y the head of the edge. The edge is said to join x and y and to be incident on x and on y. A vertex may exist in a graph and not belong to an edge. The edge (y, x) is called the inverted edge of (x, y). Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head. In one more general sense of the term allowing multiple edges, a directed graph is sometimes defined to be an ordered triple G = (V, E, ϕ) comprising:
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