Chapter 2 - Graphs and networks

What is a tree?

A connected graph with no cycles

What do we mean by 'a cycle'?

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

What do we mean by 'a Hamiltonian cycle'?

A cycle that includes every vertex

What is a Graph?

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

What do we mean by a connected graph?

A graph is connected if all it's vertices are connected

What is a complete graph?

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

What is a loop?

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

What do we mean by 'a path'?

A path is a walk in which no vertex is visited more than once

What is a planar graph?

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

What is a simple graph?

A simple graph is one in which there are no loops and there is at most one edge connecting any pair of vertices

What is a spanning tree?

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

What do we mean by 'a trail'?

A trial is a walk in which no edge is visited more than once

What do we mean by 'a walk'?

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

What do we call a graph which has weights?

A weighted graph or network

What do we call a list of edges?

Edge set

What do we mean by even degree and odd degree?

Even degree means that a vertex has an even number of incident edges. Odd degree means the vertex has an odd number of incident edges

What is an isomorphic graph?

Graphs which show the same information but may be drawn differently

What is a digraph?

If the edges of a graph have a direction associated with them they are known as directed edges and the graph is known as a directed graph often abbreviated to digraph.

What does Euler's handshaking lemma state?

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

What is an adjacency matrix?

It describes the number of arcs joining the corresponding vertices.

What is a distance matrix?

It describes the weight of each arc

What is a subgraph?

It is simply part of the original graph

What do we mean by degree/valency/order of a vertex?

The number of edges incident to it

What do we mean by weight?

The weight is a number associated with each edge in a graph.

What do we call a list of vertices?

Vertex set