# Graph Theory Glossary

This glossary is written to supplement the Interactive Tutorials in Graph Theory.  Here we define the terms that we introduce in our tutorials--you may need to go to the library to find the definitions of more advanced terms.  Please let me know of any corrections or suggestion!

Two vertices are adjacent if they are connected by an edge.

arc
A synonym for edge. See graph.

articulation point
See cut vertices.

bipartite A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V. A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has n elements and V has m, then we denote the resulting complete bipartite graph by Kn,m. The illustration shows K3,3. See also complete graph and cut vertices.

chromatic number The chromatic number of a graph is the least number of colors it takes to color its vertices so that adjacent vertices have different colors. For example, this graph has chromatic number three.
When applied to a map this is the least number of colors so necessary that countries that share nontrivial borders (borders consisting of more than single points) have different colors. See the Four Color Theorem.

circuit
A circuit is a path which ends at the vertex it begins (so a loop is an circuit of length one).

complete graph
A complete graph with n vertices (denoted Kn) is a graph with n vertices in which each vertex is connected to each of the others (with one edge between each pair of vertices). Here are the first five complete graphs: component
See connected.

connected A graph is connected if there is a path connecting every pair of vertices. A graph that is not connected can be divided into connected components (disjoint connected subgraphs). For example, this graph is made of three connected components.

cut vertex
A cut vertex is a vertex that if removed (along with all edges incident with it) produces a graph with more connected components than the original graph. See connected.

degree The degree (or valence) of a vertex is the number of edge ends at that vertex. For example, in this graph all of the vertices have degree three.

In a digraph (directed graph) the degree is usually divided into the in-degree and the out-degree (whose sum is the degree of the vertex in the underlying undirected graph).

digraph A digraph (or a directed graph) is a graph in which the edges are directed. (Formally: a digraph is a (usually finite) set of vertices V and set of ordered pairs (a,b) (where a, b are in V) called edges. The vertex a is the initial vertex of the edge and b the terminal vertex.

directed graph
See digraph.

edge
See graph.

Four Color Theorem
Every planar graph can be colored using no more than four colors.

graph
Informally, a graph is a finite set of dots called vertices (or nodes) connected by links called edges (or arcs). More formally: a simple graph is a (usually finite) set of vertices V and set of unordered pairs of distinct elements of V called edges.
Not all graphs are simple. Sometimes a pair of vertices are connected by multiple edge yielding a multigraph. At times vertices are even connected to themselves by a edge called a loop, yielding a pseudograph. Finally, edges can also be given a direction yielding a directed graph (or digraph).

in-degree
The in-degree of a vertex v is the number of edges with v as their terminal vertex. See also digraph and degree.

initial vertex
See digraph.

isolated
A vertex of degree zero (with no edges connected) is isolated.

Kuratowski's Theorem
A graph is nonplanar if and only if it contains a subgraph homeomorphic to K3,3 or K5.

length
For the length of a path see path.

loop
A loop is an edge that connects a vertex to itself. (See the illustration for degree which has a graph with three loops.) See pseudograph for a formal definition of loop.

multigraph
Informally, a multigraph is a graph with multiple edges between the same vertices. Formally: a multigraph is a set V of vertices along, a set E of edges, and a function f from E to {{u,v}|u,v in V; u,v distinct}. (The function f shows which vertices are connected by which edge.) The edges r and s are called parallel or multiple edges if f(r)=f(s). See also graph and pseudograph.

multiple edge
See multigraph.

node
A synonym for vertex. See graph.

out-degree
The out-degree of a vertex v is the number of edges with v as their initial vertex. See also digraph and degree.

parallel edge
See multigraph. path
A path is a sequence of consecutive edges in a graph and the length of the path is the number of edges traversed. (This illustration shows a path of length four.)

pendant
A vertex of degree one (with only one edge connected) is a pendant edge.

planar
A graph is planar if it can be drawn on a plane so that the edges intersect only at the vertices. (For example, of the five first complete graphs all but the fifth, K5, is planar.)

pseudograph
Informally, a pseudograph is a graph with multiple edges (or loops) between the same vertices (or the same vertex). Formally: a pseudograph is a set V of vertices along, a set E of edges, and a function f from E to {{u,v}|u,v in V}. (The function f shows which vertices are connected by which edge.) An edge is a loop if f(e) = {u} for some vertex u in V. See also graph and multigraph.

terminal vertex
See digraph.

undirected edge
Edges in graphs are undirected (as opposed to those in digraphs).

valence
See degree.

vertex
See graph.

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Chris Caldwell caldwell@utm.edu