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Discrete Mathematics Lecture 13_14: Graph Theory and Tree
By: Nur Uddin, Ph.D
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Definition of graph
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Basic terminology
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The Handshaking Theorem
Example:
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Complete Graphs
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Isomorphic Sometimes, two graphs have exactly the same form, in the sense that there is a one-to-one correspondence between their vertex sets that preserves edges. In such a case, we say that the two graphs are isomorphic. One way to represent a graph without multiple edges is to list all the edges of this graph. Another way to represent a graph with no multiple edges is to use adjacency lists, which specify the vertices that are adjacent to each vertex of the graph.
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Example: Adjecency list
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Adjacency matrix
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Incidence Matrix
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Isomorphic
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Isomorphic Isomorphic simple graphs also must have the same number of edges, because the one-to-one correspondence between vertices establishes a one-to-one correspondence between edges.
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Isomorphic evaluation using matrix
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Connectivity A path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.
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Example
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Connected components
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Euler and Hamilton Paths
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Theorems
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Example
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Hamilton Paths and Circuits
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Hamilton Puzzle
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Example
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Shortest path problem
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Exercise Determine the shortest distance form a to z.
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Dijkstra’s Algorithm
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