Edexcel A Level Maths D1 Graph Definitions

Directed graph (digraph)

When all the edges of a graph have a direction associated with them.

Even/Odd Degree

When the degree of a vertex is even/odd

Directed edges

When the edges have a direction associated with them.

A complete graph

A graph in which every vertex is directly connected by a single edge to each of the other vertices.

A simple graph

A graph in which there are no loops and there is at most one edge connecting any pair of vertices (no doubles).

Connected graph

A graph is connected if all its verticies are connected (There is a path between every vertex to every other vertex)

Planar graph

A graph that can be drawn in a plane such that no two edges meet except at a vertex.

Eulerian

A graph that contains a trail that includes every edge and starts and ends at the same vertex. Any connected graph with only even vertices is Eulerian.

Semi-Eulerian

A graph that contains a trail that includes every edge but starts and finishes at different nodes. Any connected graph with exactly 2 odd vertices is semi-Eulerian.

Weighted graph/Network

A graph that has a number associated with each edge (usually called its weight)

A subgraph

A graph, each of whose vertices belongs to the original graph and each of whose edges belong to the original graph. It is part of the original graph.

A loop

A loop is an edge that starts and finishes at the same vertex

A walk

A route through a graph along edges from one vertex to the next

A spanning tree

A subgraph of the original graph that includes all the vertices of the original graph and is also a tree.

A trail

A walk in which no edge is visted more than once

A path

A walk in which no vertex is visited more than once

A cycle

A walk in which the end vertex is the same as the start vertex and no other vertex is visited more than once.

Adjacency matrix

Each entry in the adjacency matrix descirbes the number of arcs joining the corresponding verticies.

Distance Matrix

The entries represent the weight of each arc

A graph

A graph consists of points (called verticies or nodes) which are connected by lines (edges or arcs).

A tree

A connected graph with no cycles

Hamiltonian Cycle

A cycle that includes every vertex

Isomorphic graphs

Graphs which show the same information but may be drawn differently.

Euler's Handshaking Lemma

In any undirected graph, the sum of the degrees of the verticies is equal to 2 x the number of edges. As a consequence the number of odd nodes must be even.

Degree/Valency/Order

The number of edges incident to it

Planarity algorithm

This may be applied to any graph that contains a Hamiltonian cycle. It provides a method of redrawing the graph in such a way that it becomes clear whether or not it is planar.

Conncected Verticies

Two verticies are connected if there is a path between them