An edge connects two vertices; these two vertices are said to be incident to the edge. The valency (or degree) of a vertex is the number of edges incident to it, with loops being counted twice. In the example graph vertices 1 and 3 have a valency of 2, vertices 2,4 and 5 have a valency of 3 and vertex 6 has a valency of 1. If E is finite, then the total valency of the vertices is equal to twice the number of edges. In a digraph, we distinguish the out degree (=the number of edges leaving a vertex) and the in degree (=the number of edges entering a vertex). The degree of a vertex is equal to the sum of the out degree and the in degree.
Two vertices are considered adjacent if an edge exists between them. In the above graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors for a vertex consists of all vertices adjacent to it. In the example graph, vertex 1 has two neighbors: vertex 2 and node 5. For a simple graph, the number of neighbors that a vertex has coincides with its valency.
In computers, a finite directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.
A path is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex. A path is considered simple if none of the vertices in the path are repeated. Two paths are independent if they do not have any vertex in common, except the first and last one.
The length of a path is the number of edges that the path uses, counting multiple edges multiple times. In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.
A weighted graph associates a value (weight) with every edge in the graph. The weight of a path in a weighted graph is the sum of the weights of the traversed edges. Sometimes the word cost is used instead of weight.
If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected. If it is always possible to establish a path from any vertex to any other vertex even after removing k-1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it contains k independent paths between any two vertices. The example graph above is connected (and therefore 1-connected), but not 2-connected.
A cycle (or circuit) is a path that begins and ends with the same vertex. Cycles of length 1 are loops. In the example graph, (1, 2, 3, 4, 5, 2, 1) is a cycle of length 6. A simple cycle is a cycle which has length at least 3 and in which the beginning vertex only appears once more, as the ending vertex, and the other vertices appear only once. In the above graph (1, 5, 2, 1) is a simple cycle. A graph is called acyclic if it contains no simple cycles.
An articulation point is a vertex whose removal disconnects a graph. A bridge is an edge whose removal disconnects a graph. A biconnected component is a maximal set of edges such that any two edges in the set lie on a common simple cycle. The girth of a graph is the length of the shortest simple cycle in the graph. The girth of an acyclic graph is defined to be infinity.
A tree is a connected acyclic simple graph. Sometimes, one vertex of the tree is distinguished, and called the root. Trees are commonly used as data structures in computer science (see tree data structure).
A forest is a set of trees; equivalently, a forest is any acyclic graph.
A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.
A spanning subgraph of a graph G is a subgraph with the same vertex set as G. A spanning tree is a spanning subgraph that is a tree. Every graph has a spanning tree.
A complete graph is a simple graph in which every vertex is adjacent to every other vertex. The example graph is not complete. The complete graph on n vertices is often denoted by Kn. It has n(n-1)/2 edges (corresponding to all possible choices of pairs of vertices).
A regular graph has all vertices of the same valency.
A universal graph in a class K of graphs is a simple graph in which every element in K can be embbeded as a subgraph.
A planar graph is one which can be drawn in the plane without any two edges intersecting. The example graph is planar; the complete graph on n vertices, for n> 4, is not planar.
An Eulerian path in a graph is a path that uses each edge precisely once. If such a path exists, the graph is called traversable. An Eulerian cycle is a cycle with uses each edge precisely once.
There is a dual to the Eulerian path/cycle concept. A Hamiltonian path in a graph is a path that visits each vertex once and only once; and a Hamiltonian cycle is a cycle which visits each vertex once and only once.
The example graph does not contain an Eulerian path, but it does contain a Hamiltonian path.
An empty graph is the graph whose edge set is empty.
The null graph is the graph whose edge set and vertex set are empty.
An independent set in a graph is a set of pairwise nonadjacent vertices. In the example above, vertices 1,3, and 6 form an independent set and 3,5, and 6 are another independent set.
A clique (pronounced "click") in a graph is a set of pairwise adjacent vertices. In the example graph above, vertices 1, 2 and 5 form a clique.
A bipartite graph is any graph whose vertices can be divided into two sets, such that there are no edges between vertices of the same set. A graph can be proved bipartite if there do not exist any circuits of odd length.
A k-partite graph or k-colorable graph is a graph whose vertices can be partitioned into k disjoint subsets such that there are no edges between vertices in the same subset. A 2-partite graph is the same as a bipartite graph.
A tournament is a directed graph in which each pair of vertices is connected by exactly one arc.
See also: Graph (mathematics), Graph theory, List of graph theory topics