Lecture 1_Graph_1

# Lecture 1_Graph_1 - Introduction to Graph Tommy W S Chow...

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Introduction to Graph Tommy W S Chow January 2011 Reference book: Introduction to Graph Theory by Gary Chartrand, and Ping Zhang, McGraw Hill

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Graph-based representations l Representing a problem as a graph can provide a different point of view l Representing a problem as a graph can make a problem much simpler l More accurately, it can provide the appropriate tools for solving the problem l Typical applications: logistic arrangement, path planning, and most IT applications
Bridges of Königsberg (Now Northern Germany) l Is it possible to cross all of the bridges in the city without crossing a single bridge twice? l Euler was the first mathematician to solve this problem by developing a totally new type of mathematics, GRAPH Leonhard Euler (1707-1783) (rated the 4th greatest mathematician of mankind) solved this problem

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What is graph theory? l Graph theory provides a set of techniques for analysing graphs l Complex systems graph theory provides techniques for analysing structure in a system of interacting agents, represented as a graph l In the US, they now believe graph theory would be the major mathematics to described the 21st centaury complex networking world, i.e., internet , anti-terrorist , communication networks , social network , virus spreading , and subsequent crimes. A
Bridges of Königsberg l Does this graph have a path covering every edge without duplicates? (Eulerian trail) l Theorem: A nontrivial connected graph G is Eulerian if and only if every vertex has even degree

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Friends of friends l Social experiments have demonstrated that the world is a small place after all l There is a high probability of you having an indirect connection, through a small number of friends, to a total stranger l In fact, it is postulated that a connection can be drawn between two random people in a very small number (<6 ) of links
Friends of friends l A social network l Small-world networks

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What is a graph? l A graph consists of a set of nodes and a set of edges that connect the nodes l The number of nodes or vertex is called the order of a graph Nod e Nod e Edg e Grap h
Hamiltonian Graph l A museum with 15 exhibition rooms l A security enters the reception room and checks each room once l Can he visit each room just once and return to the reception room? l From graph theory: Does the graph G have a cycle that contains every vertex? In this case, YES. l

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## This note was uploaded on 04/14/2011 for the course EE 4146 taught by Professor Tommychow during the Spring '11 term at City University of Hong Kong.

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Lecture 1_Graph_1 - Introduction to Graph Tommy W S Chow...

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