cse 321 discrete structures winter 2008 lecture 25 graph theory
Post on 21-Dec-2015
231 views
TRANSCRIPT
![Page 1: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/1.jpg)
CSE 321 Discrete Structures
Winter 2008
Lecture 25
Graph Theory
![Page 2: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/2.jpg)
Graph Theory
• Graph formalism– G = (V, E)– Vertices– Edges
• Directed Graph– Edges ordered pairs
• Undirected Graph– Edges sets of size two
![Page 3: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/3.jpg)
Graph examples
• Communication Networks
• Road networks
![Page 4: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/4.jpg)
Social networks
• Community Graph– Linked In, Face Book
• Transactions– Ebay
• Authorship– Erdos Number
![Page 5: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/5.jpg)
The web graph
![Page 6: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/6.jpg)
Page Rank
• Determine the value of a page based on link analysis
• Model of randomly traversing a graph– Weighting factors on
nodes– Damping (random
transitions)
![Page 7: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/7.jpg)
Graph terminology
• Neighborhood • Degree
![Page 8: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/8.jpg)
Degree sequence
• Find a graph with degree sequence – 3, 3, 2, 1, 1
• Find a graph with degree sequence– 3, 3, 3, 3, 3
![Page 9: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/9.jpg)
Handshake Theorem
![Page 10: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/10.jpg)
Directed Degree Theorem
![Page 11: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/11.jpg)
Special Graphs
• Complete Graphs Kn
• Cycle Cn
• Hypercube Qn
• Mesh Mn,m
![Page 12: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/12.jpg)
Bipartite Graphs
![Page 13: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/13.jpg)
2-coloring
• A graph is two colorable iff all cycles have even length
![Page 14: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/14.jpg)
Graph Representations
• Adjacency Lists
• Adjacency Matrices
• Incidence Matrices
![Page 15: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/15.jpg)
Graph Connectivity
![Page 16: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/16.jpg)
Strong connectivity vs. Weak Connectivity
![Page 17: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/17.jpg)
Strongly Connected Components
![Page 18: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/18.jpg)
Counting Paths
Let A be the Adjacency Matrix. What is A2?
d
c
b
e
a
![Page 19: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/19.jpg)
Graph Isomorphism I
Are these two graphs the same?
a d
cb
w x
y
z
![Page 20: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/20.jpg)
Graph Isomorphism II
Are these graphs the same?
![Page 21: CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory](https://reader036.vdocument.in/reader036/viewer/2022062407/56649d5f5503460f94a3f1d4/html5/thumbnails/21.jpg)
Graph Isomorphism III
Are these graphs the same?