Decision Maths Graphs Graphs Wiltshire n A graph

Decision Maths Graphs

Graphs Wiltshire n A graph is just a diagram made up of “dots” and “lines”. These are all graphs. n The dots are called “nodes” or “vertices” (singular is vertex) n The lines are called “edges” or “arcs”

Definitions 1 n Wiltshire An edge with the same vertex at each end is called a loop. n The degree or order of a vertex is the number of edges incident on it. n Question – For any graph the total of the orders of its verticies is even, why? n A simple graph is one in which there are no loops, and in which there is no more than one edge connecting any pair of vertices.

Definitions 3 n A walk is a sequence of edges in which the end of one edge (except the last) is the beginning of the next. Wiltshire n A trail is a walk in which no edge is repeated.

Definitions 4 Wiltshire n A path is a trail in which no vertex is repeated. n A graph is connected if there exists a path between every pair of vertices.

Definitions 5 Wiltshire n A cycle is a closed path if the end of the last edge is the start of the first. n A Hamiltonian cycle is a cycle which visits every vertex once and only once.

Definitions 6 Wiltshire n A tree is a simple connected graph with no cycles. A tree Not trees

Definitions 7 Wiltshire n A Digraph (Directed Graph) is a graph in which at least one edge has a direction associated with it. n A complete graph is a simple graph in which every pair of vertices is connected by an edge.

Definitions 8 Wiltshire n An incidence matrix is a way of representing the number of edges between nodes in a matrix. The graph below is represented by the matrix next to it.

Definitions 9 Wiltshire n Two graphs are Isomorphic if one can be stretched, twisted or otherwise distorted into the other. n Which two graphs below are Isomorphic? n If two graphs are isomorphic then the labels on them must correspond to each other.

Definitions 10 Wiltshire n A planar graph is one which can be drawn without any edges crossing. n Which graph(s) below is Planar? n Draw two examples of Planar graphs.

Definitions 11 Wiltshire n A bipartite graph is one in which the vertices fall into two sets and in which each edge has a vertex from one set at one end and from the other set at its other end.

Question 1 Wiltshire n X = { 2, 3, 4, 5, 6} Draw a graph to represent the relationship ‘share a common factor other than 1’

Question 2 Wiltshire n X = { London, Oxford, Birmingham, Cambridge, Leicester} n Let X x X be the set of all possible pairs from the set X. (there exists a road between the two towns) n Xx. X= { (London, Oxford) (London, Birmingham) (London, Cambridge) (London, Leicester) (Oxford, London) (Oxford, Birmingham) (Birmingham, London) (Birmingham, Oxford) (Birmingham, Leicester) (Cambridge, London) (Leicester, Birmingham) } n Draw a graph to show the set X x X.

Question 2 Wiltshire

Question 3 – Ex 2 A q 13 pg 54 n Each node represents a section of land. n And each arc is the route over the bridges. Wiltshire

Eulerian Wiltshire n A graph is called Eulerian or traversable if each can be traced once and only once, without lifting pencil from paper. n A graph is traversable if it has no odd vertices or just two odd vertices. n Prove that the graph below is traversable.