Graph Theory and Its Applications

Graph Theory

N O M A N I S L A M

Outline

 Introduction

 Graph Models in Communication and Operating Systems

 Representing Graphs in Computers

 Graph Coloring

 Spanning Trees

 Other Problems in Graph Theory

Graph

 A graph G(V,E) is mathematical

structure usually represented as a diagram consisting of

points/nodes/vertices V joined by lines/edges E

 Vertices may correspond to chemical

elements, towns, terminals)

 Edges may correspond to chemical

bonds, roads, wires

 Graph theory has applications in

Terminologies

 directed & undirected

graph

 _{simple and multigraph}  _{vertex and edge}

adjacency

 _{in and out-degree}

(valency)

 _{bipartite graph}  path, circuit, loop  _{connected graph}  _{complete graph K}

n and

Terminologies (contd)

 weighted graph

 Euler cycle and

Hamiltonean cycle

 tree and forest

 spanning tree

Weighted Graph with Hamiltonean Cycle

A,B,C,D,E,A

Spanning Trees of K

Euler Cycle

Graph in Communication

Interconnection network for five processors (a complete graph)

Graph in Operating Systems

New Ready Running

Exit

Blocked

Admit Dispatch

Timeout Release

Event

Occurs _Event

Wait

Graph in Operating Systems

0 1 2 3 4

5

Eulerian and Hamiltonean Type

Problems



Eulerian Type Problems

– Diagrams Tracing Puzzles – Mazes



Hamiltonean Type Problems

– Gray Codes

– Traveling Sales Man Problem

 A sales person wants to minimize the total traveling cost

to visit all the cities in a territory and return to the starting point. Other applications include routing trucks for

Exercise

 Draw the digraphs whose vertices and arcs are as follows.

In each case say if the digraph is a simple digraph.

– a) V = {u, v, w, x}, A = {vw, wu, wv, wx, xu}

– _{b) V = {1, 2, 3, 4, 5, 6, 7, 8}, A = {12, 22, 23, 24, 34, 35, 67,}

68, 78}

– c) V = {n, p, q, r, s, t}, A = {np, nq, nt, rs, rt, st, pq, tp}

 _{Which of the following graphs are Eulerian and/or}

Representing Graphs

 Adjacency Matrix

a_ij = 1 if {v_i,v_j} is an edge of G 0 otherwise

 Incidence Matrix

m_ij = 1 if edge e_J is incident with v_i 0 otherwise

0 1 0 1

1 0 1 1

0 1 0 0

1 1 1 0

1 2

4 3

1 0 0 1 0

1 1 0 0 1

0 1 1 0 0

0 0 1 1 1

1

4 5 2

3

Exercise

 Write down the

adjacency and

incidence matrices of the following graphs and digraphs:

 Draw the digraphs

Exercise



How many edges are there in a complete

graph on 5 vertices?



How many edges are there in general in a

Shortest Path Algorithm



In a weighted graph, what is the shortest

path between two given vertices?



In computer network for example, what are

the set of telephone lines that for connecting

Karachi and London gives the least cost or

the fastest communication (Circuit Switching)

Graph Coloring



Assignment of colors to each vertex of graph

so that adjacent vertices have different colors



Least number of colors needed to color a

graph is called chromatic number

Graph Coloring Applications



Scheduling Final Exams

– How can the final exams be scheduled so that no

students has two exams at the same time

– The nodes of graph corresponds to the courses

and there will be links between two nodes if the exam of the two courses can’t be taken

simultaneously

Graph Coloring Applications



Frequency Assignments

– Assigning frequencies to mobile radios and other

users of the electromagnetic spectrum

– Two customers that are sufficiently close must

be assigned different frequencies, while those that are distant can share frequencies

– The nodes of graphs are mobile and there will be

links between two nodes if they can’t use the same frequencies

– The problem of minimizing the number of

Graph Coloring Applications



Register Allocation

– Assign variables of a program to a limited number

of hardware registers during program execution

– The goal is to assign variables that do not conflict

so as to minimize the use of non-register memory.

– Create a graph where the nodes represent

Graph Coloring Applications

 Aircraft Scheduling

– We have k aircrafts and we have to assign n-flights

– _{If two flights overlap, we can’t assign same aircraft to both}

flights

– Vertices of graph corresponds to flights, two vertices are

connected if corresponding time intervals overlap

– _{Assignment of colors means assignment of aircrafts}

– Graph Coloring will then try to minimize the number of

Graph Coloring Algorithms



Graph Coloring is NP-Complete problem



Sequential Coloring – Always colors the next

vertex with the minimum acceptable color

(i.e. the minimum color not already assigned

to a vertex adjacent to v

)



Sequential Color Interchange

Spanning Tree



A sub-graph of a graph that is a tree

containing every vertex of G

Spanning Tree Applications



IP Multicasting

– Unicasting, Multicasting, Broadcasting

– Multicasting means to send data from on

computer to a set of other computers

– Instead of unicasting to each source individually,

Exercise



Find all the spanning trees in each of the two

graphs shown:



Prove that any tree is bipartite



Show that any tree can be colored with two

Some Other Graph Problems

 Vertex Cover Problem – A subset of vertices such

that each edge is incident upon some vertex in C

(what is the minimum number of sentries that can be posted at intersections such that all corridors are

monitored)

 Clique Problem – A subset of vertices K such that

every pair of distinct vertices in K is joined by an edge of G

Some Other Graph Problems

C

C C

I

I

I

K

K

K

Vertex Cover Independent Set

References

 URLs

– ‘Graph Theory Glossary’, http://www.utm.edu

– _{‘Graph Theory Glossary’, http://exchange.manifold.net}

– ‘Graph coloring applications’, http://mat.gsia.cmu.edu

/COLOR

– http://www.personal.kent.edu/~rmuhamma/GraphTheory/gr

aphTheory.htm

Relevant Questions