Basic Theory of Graph (Part 1)
Some Formal De…nitions of Graph – Matrix Representation of Graph
MZI
School of Computing
Telkom University
SoC Tel-U
April-May 2023
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Basic Theory of Graph (Part 1)
Some Formal DeÖnitions of Graph ñ Matrix Representation of Graph
MZI
School of Computing Telkom University
SoC Tel-U
April-May 2023
Acknowledgements
This slide is composed based on the following materials: (^1) Discrete Mathematics and Its Applications, 8th Edition, 2019, by K. H. Rosen (main). (^2) Discrete Mathematics with Applications, 5th Edition, 2018, by S. S. Epp. (^3) Mathematics for Computer Science. MIT, 2010, by E. Lehman, F. T. Leighton, A. R. Meyer. (^4) Slide for Matematika Diskret 2 (2012). Fasilkom UI, by B. H. Widjaja. (^5) Slide for Matematika Diskret 2 at Fasilkom UI by Team of Lecturers. (^6) Slide for Matematika Diskret. Telkom University, by B. Purnama. Some of the pictures are taken from the above resources. This slide is intended for academic purpose at FIF Telkom University. If you have any suggestions/comments/questions related with the material on this slide, send an email to
Contents
(^1) Background and Motivation
(^2) Some Formal DeÖnitions of Graph
(^3) Some Basic Terminologies
4 Subgraph, Spanning Subgraph, Complement Graph, and Graph Union
5 Some Simple Graphs with Special Structure
(^6) Graph representation with Matrix and List
Background
Graph is an important object in Discrete Math and has many implementations, one of them is in topology design of communication networks.
We can use graph to model the connectedness between discrete objects. One of them is a graph that describe connectedness between cities in Central Java (here, we view connectedness from the availability of the road connecting the cities).
Motivation and Informal Terminologies
By modelling connectedness of cities in Central Java using graph, we can answer the following questions:
Motivation and Informal Terminologies
By modelling connectedness of cities in Central Java using graph, we can answer the following questions: (^1) What is the shortest route that connect Pekalongan and Solo?
Motivation and Informal Terminologies
By modelling connectedness of cities in Central Java using graph, we can answer the following questions: (^1) What is the shortest route that connect Pekalongan and Solo? (^2) Can we visit every city in Central Java and pass through the cities exactly once in one journey? (^3) How many di§erent routes that can be used by a person from Cilacap to Rembang if the number of cities that is passed through must be as minimum as possible?
Motivation and Informal Terminologies
By modelling connectedness of cities in Central Java using graph, we can answer the following questions: (^1) What is the shortest route that connect Pekalongan and Solo? (^2) Can we visit every city in Central Java and pass through the cities exactly once in one journey? (^3) How many di§erent routes that can be used by a person from Cilacap to Rembang if the number of cities that is passed through must be as minimum as possible? By graph modeling, cities are viewed as dot or node or vertex (plural: vertices) while roads are viewed as edge or line or arc.
Contents
(^1) Background and Motivation
(^2) Some Formal DeÖnitions of Graph
(^3) Some Basic Terminologies
4 Subgraph, Spanning Subgraph, Complement Graph, and Graph Union
5 Some Simple Graphs with Special Structure
(^6) Graph representation with Matrix and List
Directed Graph with Multiple Edges
DeÖnition (a directed graph with multiple edges)
A graph G is denoted as a triple (V; E; f ) where (^1) V is a set of all vertices in the graph, (^2) E is a set of all edges in the graph, (^3) f is a total function from E to V V.
A directed graph that has multiple edges as well as loop is called arbitrary directed graph or directed multigraph.
Solution of Exercise 1
We have G = (V; E; f ) where (^1) V =
Solution of Exercise 1
We have G = (V; E; f ) where (^1) V = f 1 ; 2 ; 3 ; 4 g, (^2) E =
Solution of Exercise 1
We have G = (V; E; f ) where (^1) V = f 1 ; 2 ; 3 ; 4 g, (^2) E = fe 1 ; e 2 ; e 3 ; e 4 ; e 5 ; e 6 g, (^3) f : E! V V with the deÖnition: I (^) f (e 1 ) = f (e 2 ) = (1; 2) I (^) f (e 3 ) = (3; 4) I (^) f (e 4 ) = (4; 3) I (^) f (e 5 ) = f (e 6 ) = (4; 4).
Undirected Graph with Multiple Edges
DeÖnition (an undirected graph with multiple edges)
A graph G is denoted as a triple (V; E; f ) where (^1) V is a set of all vertices in the graph, (^2) E is a set of all edges in the graph, (^3) f is a total function from E to a set ffu; vg j u; v 2 V g.
An undirected graph that has multiple edges as well as loop is called a pseudograph.