1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using

1

GRAPH Learning Outcomes Students should be able to:Explain basic terminology of a graphIdentify Euler and Hamiltonian cycleRepresent graphs using adjacency

matrices

2

Contents Introduction Paths and circuits Matrix representations of graphs

3

Introduction to Graphs

DEF: A simple graph G = (V,E ) consists of a non-empty set V of vertices (or nodes) and a set E (possibly empty) of edges where each edge is associated with a set consisting of either one or two vertices called its endpoints.

Q: For a set V with n elements, how many possible edges there?

4

Terminology Loop, parallel edges, isolated, adjacent Loop - an edge connects a vertex to

itself Two edges connect the same pair of

vertices are said to be parallel. Isolated vertex – unconnected vertex. Two vertices that are connected by an

edge are called adjacent. An edge is said to be incident on each

of its end points.

5

Example of a graph Vertex set = {u1, u2, u3}

Edge set = {e1, e2, e3, e4}

e1, e2, e3 are incident on u1

u2 and u3 are adjacent to u1

e4 is a loop

e2 and e3 are parallel

6April 19, 2023 Graphs and Trees 6

Types of Graphs

Directed – order counts when discussing edges

Undirected (bidirectional)

Weighted – each edge has a value associated with it

Unweighted


Examples

http://richard.jones.name/google-hacks/google-cartography/google-cartography.html


Special Graphs Simple – does not have any loops or

parallel edges Complete graphs – there is an edge

“between” every possible tuple of vertices Bipartite graph – V can be partitioned into

V1 and V2, such that: – (x,y)E (xV1 yV2) (xV2 yV1)

Sub graphs– G1 is a subset of G2 iff

Every vertex in G1 is in G2 Every edge in G1 is in G2

Connected graph – can get from any vertex to another via edges in the graph

9

Complete Graphs there is an edge “between” every

possible tuple of vertices. |e| = C(n,2) = n. (n-1)/2

10

Bipartite graph 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.

11

Complete bipartite A bipartite graph is a complete bipartite

graph if every vertex in U is connected to every vertex in V. If U has m elements and V has n, then we denote the resulting complete bipartite graph by Km,n. The illustration shows K3,2


Degree of Vertex Defined as the number of edges attached

(incident) to the vertex. A loop is counted twice.


Handshake Theorem If G is any graph, then the sum of the

degrees of all the vertices of G equals twice the number of edges of G.

Specifically, if the vertices of G are v1, v2, …, vn, where n is a nonnegative integer, then:– The total degree of G = d(v1)+d(v2)+…+d(vn)

= 2 (the number of edges of G)


Total degree of a graph is even

Prove that the total of the degrees of all vertices in a graph is even.

Since the total degree equals 2 times of edges, which is an integer, the sum of all degree is even.

15

Whether certain graphs exist

Draw a graph with the specified properties or show that no such graph exists.

(a) A graph with four vertices of degrees 1,1,2, and 3

(b) A graph with four vertices of degrees 1,1,3 and 3

(c) A simple graph with four vertices of degrees 1,1,3 and 3


Even no. of vertices with odd degree

In any graph, there are an even number of vertices with odd degree

Is there a graph with ten vertices of degrees 1,1,2,2,2,3,4,4,4, and 6?

17

Learning Outcomes Students should be able to:Explain basic terminology of a graphIdentify Euler and Hamiltonian cycleRepresent graphs using adjacency

matrices


Seven Bridges of Königsberg

Is it possible for a person to take a walk around town, starting and ending at the same location and crossing each of the seven bridges exactly once?

No


Definitions Terminology - Walk, path, simple path,

circuit, simple circuit. Walk from two vertices is a finite

alternating sequence of adjacent vertices and edges– Trivial walk from v to v consists of single

vertex

v0e1v1e2…envn

20

Path Path – a walk that does not contain a

repeated edge (may have a repeated vertex)

v0e1v1e2…envn where all the ei are distinct

Simple path – a path that does not contain a repeated vertex

v0e1v1e2…envn where all the ei and vj are distinct

21

Circuit Closed walk – starts and ends at

same vertex Circuit – a closed walk without

repeated edge Simple circuit – a circuit with no

repeated vertex except first and last

22

Connectedness Connectedness – if there is a walk

from one to the other


Euler Circuits A circuit that contains every vertex

and every edge of G. A sequence of adjacent vertices and

edges– that starts and ends at the same vertex, – uses every vertex of G at least once,

and– uses every edge of G exactly once.


If a graph has an Euler circuit, every vertex has even degree.

Contrapositive: if some vertex has odd degree, then the graph does not have an Euler circuit.


If every vertex of nonempty graph has even degree and if graph is connected, then the

graph has an Euler circuit.


Euler Circuit A graph G has an Euler circuit if, and

only if, G is connected and every vertex of G has even degree.


Hamiltonian PathA path in an undirected graph which visits each vertex exactly once.


Hamiltonian Circuit A simple circuit that includes every

vertex of G. A sequence of adjacent vertices and

distinct edges in which every vertex of G appears exactly once, except for the first and last, which are the same.


Hamiltonian Circuit Proved simple criterion for

determining whether a graph has an Euler circuit

No analogous criterion for determining whether a graph has a Hamiltonian circuit

Nor is there an efficient algorithm for finding such an algorithm


Traveling Salesman Problem

http://en.wikipedia.org/wiki/Traveling_Salesman_Problem


TSP One way to solve the general problem is to:

– Write down all Hamiltonian circuits– Compute total distance for each– Pick one for which total is minimal

What if graph has 30 vertices:– 29! =8.84 x 1030 different Hamiltonian circuits– If each circuit could be found and total distance

computed in a nanosecond, then would take: 2.8 x 1014 years!!!

– No known algorithm that is more efficient!!!– Some that find “pretty good” solutions

32

Learning Outcomes

Students should be able to:Explain basic terminology of a graphIdentify Euler and Hamiltonian cycleRepresent graphs using adjacency

matrices


Matrix Representations of Graphs


Matrices and Connected Components


Counting Walks of Length n Matrix multiplication


How do these graphs relate?

=

37

Summary

Documents

1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using