Graph Definitions

Graph
A visual representation of a set of vertices (nodes). Each member of X is represented as a vertex. Connections are represented by edges (arcs).
Vertex / Node
A point representing a member of the set.
Edge / Arc
A link between vertices. An edge must have a vertex at each end.
Loop
An edge with the same vertex at each end.
The basic parts of a graph: edges, vertices, and a loop

Figure 1: Edges, vertices, and a loop, the basic parts of a graph.

Degree (order) of a vertex
The number of edges meeting at the vertex.
Simple graph
Contains no loops and has no more than one edge between any pair of vertices.
Walk
Sequence of edges. The end of each edge (except the last) is the start of the next.
Trail
A walk with no repeated edges.
Path
A trail in which no vertex is repeated.
Cycle
A closed path. The end of the last edge joins the start of the first.
Hamiltonian Cycle
A cycle that visits every vertex
Connected
There is a path between every pair of vertices in a connected graph.
Tree
Simple connected graph with no cycles.
Digraph (directed graph)
Graph with at least one edge having a direction.
Complete graph
A simple graph with each pair of vertices connected by an edge.
Eulerian
An Eulerian graph is connected and the degree of every vertex is even.
Incidence Matrix
Matrix representation of a graph. Elements of the matrix show the number of edges connecting the vertices represented by the row and column of the matrix.
Isomorphic
Graphs are isomorphic if one can be deformed to make the other. The vertices can be moved and the edges straightened or bent to do this.
Planar graph
A graph that can be drawn such that no edges cross.
Bipartite graph (bigraph)
A graph joining two sets. Every edge starts in one set and ends in the other.