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Unformatted text preview: Discrete Mathematics Discrete Mathematics (2009 Spring) Graphs (Chapter 9, 5 hours) ChihWei Yi Dept. of Computer Science National Chiao Tung University June 1, 2009 Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models What are Graphs? General meaning in everyday math: A plot or chart of numerical data using a coordinate system. Not Not Technical meaning in discrete mathematics: A particular class of discrete structures (to be de&ned) that is useful for representing relations and has a convenient webbylooking graphical representation. Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Simple Graphs Correspond to symmetric binary relations R . Visual Representation of a Simple Graph Visual Representation of a Simple Graph A simple graph G = ( V , E ) consists of: 1 A set V of vertices or nodes ( V corresponds to the universe of the relation R ). 2 A set E of edges / arcs / links: unordered pairs of [distinct?] elements u , v 2 V , such that uRv . Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Example of a Simple Graph Let V be the set of states in the farsoutheastern U.S.: V = f FL,GA,AL,MS,LA,SC,TN,NC g . Let E = ff u , v g j u adjoins v g = 8 < : {FL,GA},{FL,AL},{FL,MS},{FL,LA},{GA,AL}, {AL,MS},{MS,LA},{GA,SC},{GA,TN},{SC,NC}, {NC,TN},{MS,TN},{MS,AL} 9 = ; . TN AL MS LA SC GA FL NC TN AL MS LA SC GA FL NC Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Multigraphs Like simple graphs, but there may be more than one edge connecting two given nodes. A multigraph G = ( V , E , f ) consists of a set V of vertices, a set E of edges (as primitive objects), and a function f : E ! ff u , v g j u , v 2 V ^ u 6 = v gg . E.g., nodes are cities, edges are segments of major highways. Parallel edges Parallel edges Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Pseudographs Like a multigraph, but edges connecting a node to itself are allowed. A pseudograph G = ( V , E , f ) where f : E ! ff u , v g j u , v 2 V g . Edge e 2 E is a loop if f ( e ) = f u , u g = f u g . E.g., nodes are campsites in a state park, edges are hiking trails through the woods. Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Directed Graphs Correspond to arbitrary binary relations R , which need not be symmetric. A directed graph ( V , E ) consists of a set of vertices V and a binary relation E on V . E.g.: V = people, E = f ( x , y ) j x loves y g . Discrete Mathematics Chapter 9 Graphs §9.1 Graphs and Graph Models Directed Multigraphs Like directed graphs, but there may be more than one arc from a node to another. A directed multigraph G = ( V , E , f ) consists of a set V of vertices, a set E of edges, and a function f : E ! V & V ....
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This note was uploaded on 10/25/2009 for the course EE 2011 taught by Professor Denny during the Spring '09 term at National Tsing Hua University, China.
 Spring '09
 Denny

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