Graphs and Networks

Equation for the number of edges

1/2*n(n-1)

Complete bipartite graph

A bipartite graph in which every possible edge is present, Kmn

Walk

A continuous sequence of edges. The end vertex of one edge in the sequence is the start vertex of the next

Graph

A diagram involving a set of points and interconnecting lines

Subgraph

A graph formed by some of the vertices and edges of a given graph

Directed graph / Digraph

A graph in which at least one edge has a direction associated with it

Bipartite graph

A graph in which there are two distinct sets of vertices, and each edge joins a vertex in one set to a vertex in the other

Connected

A graph is connected if it possible to travel from any vertex to any other vertex, even passing through other vertices along the way

Traversable / Eulerian

A graph is traversable or Eulerian if it has a trail which is closed (starts and ends at the same vertex) and which includes every edge once only. All of the vertices are of an even degree

Planar graph

A graph which can be drawn without crossing any edges

Network / Weighted graph

A graph with a number, or weight, associated with each edge

Simple graph

A graph with no loops or multiple edges

Cycle / Circuit / Closed walk

A journey which returns to its start vertex

Edge/Arc

A line joining two vertices

Adjacency matrix

A matrix of the connections in a graph, showing the number of edges connecting the vertices. Does not include directions

Distance matrix

A matrix of the connections in a network, including directions and weights. Entries correspond to the weights of the edges

Triangle inequality

A network satisfies the triangle inequality, for every set of three connected vertices A, B and C: weight AB <= weight AC + weight BC

Vertex/Node

A point in a graph

Tree

A simple connected graph with no cycles

Complete graph, Kn

A simple graph with an edge connecting each possible pair of vertices

Path

A walk which does not pass through any vertex more than once

Directed network

A weighted graph with directed edges

Hamiltonian Cycle

Goes through all the vertices exactly once and returns to the start

Isomorphic

Graphs which have the same set of vertices and connections

Non-traversable graph

Has more than two odd nodes

Edge set

List of edges, e.g. {AB, AC, BC}

Vertex set

List of vertices, e.g {A, B, C}

Bidirectional edges

Listed twice in edges sets/matrices because it goes both ways, e.g. CD and DC. Similarly DD is listed twice

Semi-traversable /Semi-Eulerian

The graph has a trail which includes every edge once only but it starts and finishes at two different vertices. The two odd vertices are the start and finish nodes.

Degree/Order/Valency

The number of edges which connect to a vertex

Equation for total of degrees

Total of degrees = 2 * number of edges