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![Basic Terminology
Regular Graph
▸ All the vertices have the same degree.
▸ D-regular graph:
▸ All the vertices have degree d.
▸ [Question] Draw a 3-regular undirected graph
with 8 vertices.](https://image.slidesharecdn.com/introductiontographtheoryskip-180922053843/75/Introduction-to-Graph-Theory-25-2048.jpg)
![Basic Terminology
[Question] Walk / Path / Circuit / Cycle?
▸ 1) <a, b, a>
▸ 2) <a, b, d, e, c>
▸ 3) <a, b, d, e, c, b>
▸ 4) <a, b, d, e, c, a>](https://image.slidesharecdn.com/introductiontographtheoryskip-180922053843/75/Introduction-to-Graph-Theory-32-2048.jpg)
![Graph Classes
Complete Graphs (Kn)
▸ Each vertex pair is connected by an edge.
▸ [Question] How many edges does Kn have?](https://image.slidesharecdn.com/introductiontographtheoryskip-180922053843/75/Introduction-to-Graph-Theory-36-2048.jpg)
![Historical Problems
Eulerian Trail (Eulerian Path)
▸ [Question] Find an Eulerian trail.
▸ There are effective algorithms to construct Eulerian trais.
▸ Fleury's algorithm (1883)
▸ Hierholzer's algorithm (1873)
https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/introduction-euler-paths/](https://image.slidesharecdn.com/introductiontographtheoryskip-180922053843/75/Introduction-to-Graph-Theory-44-2048.jpg)
![Historical Problems
Hamiltonian Path / Cycle
▸ Path / cycle which visits every vertex exactly once.
▸ No effective algorithm to construct Hamiltonian paths.
▸ [Question] Draw a graph that does not have any Hamiltonian paths.](https://image.slidesharecdn.com/introductiontographtheoryskip-180922053843/75/Introduction-to-Graph-Theory-45-2048.jpg)
Uploaded byYosuke Mizutani
Introduction to Graph Theory
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
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Introduction to Graph Theory
- 1. Introduction to GraphTheory Yosuke Mizutani Presents 2018/09/21@RTP Kinyo Kai
- 2.
- 3. What are graphs? Graphsin Graph Theory ▸ G = (V, E) ▸ V := Set of vertices (nodes) ▸ E := Set of edges ▸ Model relationships between pairs of objects
- 4. What are graphs? UndirectedGraphs ▸ V = {a, b, c, d, e} ▸ E = { {a, b}, {a, c}, {b, c}, {b, d}, {d, e}, {b, e} } |V| = 5 |E| = 6
- 5. What are graphs? DirectedGraphs ▸ V = {a, b, c, d, e} ▸ E = { (a, b), (a, c), (c, b), (b, d), (d, e), (e, c) } |V| = 5 |E| = 6
- 6. What are graphs? DirectedGraphs ▸ Can we traverse this graph from a to e? ▸ Can we traverse this graph from e to a?
- 7.
- 8. Examples of graphs RailMap https://www.reddit.com/r/WTF/comments/1ng9py/a_simple_map_of_the_tokyo_metro/
- 9. Examples of graphs RoadMap http://ontheworldmap.com/usa/state/north-carolina/north-carolina-highway-map.jpg
- 10. Examples of graphs NetworkDevices https://neo4j.com/business-edge/managing-network-operations-with-graphs/
- 11. Examples of graphs SocialNetwork https://www.flickr.com/photos/clintjcl/4126188232/
- 12. Examples of graphs Moreno'sSociogram http://www.martingrandjean.ch/social-network-analysis-visualization-morenos-sociograms-revisited/
- 13. Examples of graphs WorldWide Web https://en.wikipedia.org/wiki/World_Wide_Web
- 14.
- 15. Examples of graphs Physics- Feynman Diagrams https://www.quora.com/What-is-the-most-complex-Feynman-Diagram
- 16. Examples of graphs EvolutionTrees https://www.reddit.com/r/gaming/comments/5ie1cd/pokemon_tree_of_life_for_generation_7/
- 17. Examples of graphs FileSystems
- 18. Examples of graphs GameTrees https://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta.html
- 19. Examples of graphs DependencyGraphs http://euclid.nmu.edu/~mkowalcz/
- 20.
- 21.
- 22. Basic Terminology Degree ▸ Totalnumber of edges connected to a vertex ▸ Total degree of a graph: ▸ Sum of the degrees of all of the vertices.
- 23. Basic Terminology Degree ▸ Whatis the total degree of this graph? |V| = 5 |E| = 6
- 24. Basic Terminology Parallel Edges/ Self Loop ▸ Parallel Edges: ▸ Multiple edges between the same pair of vertices ▸ Self Loop: ▸ Edge between a vertex and itself
- 25. Basic Terminology Regular Graph ▸All the vertices have the same degree. ▸ D-regular graph: ▸ All the vertices have degree d. ▸ [Question] Draw a 3-regular undirected graph with 8 vertices.
- 26. Basic Terminology Regular Graph ▸3-regular graph
- 27. Basic Terminology Subgraph ▸ G'= (V', E') is a subgraph of a graph G = (V, E) if ▸ V' ⊆ V and E' ⊆ E
- 28. Basic Terminology Walk ▸ Fromv0 to vl ▸ Ex. <a, b, d, b, c> ▸ Can repeat a vertex or edge
- 29. Basic Terminology Path ▸ Fromv0 to vl ▸ Ex. <a, b, c> ▸ No vertex is repeated
- 30. Basic Terminology Circuit ▸ Fromv0 to v0 ▸ Ex. <a, b, d, e, c, b, a> ▸ Can repeat a vertex or edge
- 31. Basic Terminology Cycle ▸ Fromv0 to v0 ▸ Ex. <a, b, c, a> ▸ Length: at least three ▸ No vertex is repeated except the first and last
- 32. Basic Terminology [Question] Walk/ Path / Circuit / Cycle? ▸ 1) <a, b, a> ▸ 2) <a, b, d, e, c> ▸ 3) <a, b, d, e, c, b> ▸ 4) <a, b, d, e, c, a>
- 33.
- 34. Graph Classes Path Graphs(Pn) ▸ Two vertices of degree 1 ▸ n-2 vertices of degree 2
- 35. Graph Classes Cycle Graphs(Cn) ▸ Single cycle through all vertices.
- 36. Graph Classes Complete Graphs(Kn) ▸ Each vertex pair is connected by an edge. ▸ [Question] How many edges does Kn have?
- 37. Graph Classes Bipartite Graphs ▸Vertices: two disjoint sets. ▸ No two vertices within the same set are connected.
- 38. Graph Classes Bipartite Graphs ▸Is this a bipartite graph?
- 39. Graph Classes Grid Graphs(Gm,n) ▸ Pm×Pn G7,4
- 40. Graph Classes Hypercube Graphs(Qn) ▸ Has 2n vertices. ▸ Each vertex is labeled with an n-bit string. ▸ Two vertices are connected by an edge if their corresponding labels differ by only one bit.
- 41.
- 42. Historical Problems Seven Bridgesof Königsberg (1736) ▸ Find a walk through the city that would cross each of the bridges once and only once https://www.amusingplanet.com/2018/08/the-seven-bridges-of-konigsberg.html
- 43. Historical Problems Eulerian Trail(Eulerian Path) ▸ Trail (walk without repeated edges) which visits every edge exactly once. ▸ Eurlean trails exist if 0 or 2 vertices have an odd degree.
- 44. Historical Problems Eulerian Trail(Eulerian Path) ▸ [Question] Find an Eulerian trail. ▸ There are effective algorithms to construct Eulerian trais. ▸ Fleury's algorithm (1883) ▸ Hierholzer's algorithm (1873) https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/introduction-euler-paths/
- 45. Historical Problems Hamiltonian Path/ Cycle ▸ Path / cycle which visits every vertex exactly once. ▸ No effective algorithm to construct Hamiltonian paths. ▸ [Question] Draw a graph that does not have any Hamiltonian paths.
- 46. Historical Problems Vertex Coloring ▸Label all vertices using at most k colors. ▸ No two vertices sharing the same edge have the same color. https://en.wikipedia.org/wiki/Graph_coloring
- 47. Historical Problems Traveling SalesmanProblem (TSP) ▸ Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city? http://mathgifs.blogspot.com/2014/03/the-traveling-salesman.html
- 48. Historical Problems Traveling SalesmanProblem (TSP) ▸ What is the shortest TSP route starting from c1? ▸ How much time would you take to compute the route if the number of cities is 10 / 100 / 1000? Dr. Jason King, CSC 316 Course Material
- 49.
- 50. Complexity Analysis The RandomAccess Machine (RAM) Model ▸ A computer consists of: ▸ A CPU ▸ A bank with an unlimited number of memory cells ▸ Assumptions ▸ Accessing any cell in memory takes constant time ▸ Each "simple" operation takes 1 time step Dr. Jason King, CSC 316 Course Material
- 51. Complexity Analysis The RandomAccess Machine (RAM) Model ▸ Example: one "simple" operation takes 1 ns (10-9 seconds) ▸ How much time would it take if the number of "simple" operations are: ▸ 109 ▸ 1010 ▸ 1011
- 52. Complexity Analysis Growth ofFunctions ▸ The larger input size, the longer the computation would take. ▸ Let T(n) be the computation time of an algorithm for some input n. ▸ Changing hardware/software environment affects T(n) by a constant factor. (At least for today's computers...) Dr. Jason King, CSC 316 Course Material
- 53. Complexity Analysis Growth ofFunctions n=10 n=20 n=100 n=1000 n=106 n=109 n=1012 T(log2 n) 3 ns 4 ns 7 ns 10 ns 20 ns 30 ns 40 ns T(√n) 3 ns 4 ns 10 ns 32 ns 1 μs 32 μs 1 ms T(n) 10 ns 20 ns 100 ns 1 μs 1 ms 1 s 17 min T(n log2 n) 33 ns 86 ns 660 ns 10 μs 20 ms 30 s 11 h T(n2) 100 ns 400 ns 10 μs 1 ms 17 min 32 yrs 3×107 yrs T(n3) 1 μs 8 μs 1 ms 1 s 32 yrs 3×1010 yrs 3×1019 yrs T(2n) 1 μs 1 ms 4×1013 yrs 3×10284 yrs - - - T(n!) 4 ms 77 yrs 9×10157 yrs - - - - When T(1) = 1 ns Our universe: 1.3×1010 yrs old
- 54. Complexity Analysis Growth ofFunctions n=10 n=20 n=100 n=1000 n=106 n=109 n=1012 T(log2 n) <1 ns <1 ns <1 ns <1 ns <1 ns <1 ns <1 ns T(√n) <1 ns <1 ns <1 ns <1 ns <1 ns <1 ns <1 ns T(n) <1 ns <1 ns <1 ns <1 ns 10 ns 10 μs 10 ms T(n log2 n) <1 ns <1 ns <1 ns <1 ns 200 ns 300 μs 400 ms T(n2) <1 ns <1 ns <1 ns 10 ns 10 ms 3 h 300 yrs T(n3) <1 ns <1 ns 10 ns 10 μs 3 h 3×105 yrs 3×1014 yrs T(2n) <1 ns 10 ns 4×108 yrs 3×10279 yrs - - - T(n!) 40 ns 7 h 9×10152 yrs - - - - When we have a supercomputer and assume T(1) = 1×10-14 s Our universe: 1.3×1010 yrs old
- 55. Complexity Analysis Growth ofFunctions Dr. Jason King, CSC 316 Course Material
- 56. Complexity Analysis Big-Oh Notation ▸Defines asymptotic upper bound ▸ Given functions f(n) and g(n), we say that f(n) is O(g(n)) if there are positive constants c and n0 such that 0 ≤ f(n) ≤ c g(n) for all n ≥ n0 Dr. Jason King, CSC 316 Course Material
- 57. Complexity Analysis Big-Oh Notation ▸Example: ▸ f(n) = 3n2 + 2n + 10000 is O(n2) ▸ f(n) = n100 + 1.01n is O(1.01n) ▸ Finding Eulerian trails: O(|E|) ▸ Solving the traveling salesman problem: O(n2・2n) ▸ Categorized as NP (Non-deterministic Polynomial-time) -hard
- 58.
- 59. Centrality Analysis Degree Centrality ▸Degree of the vertex (very simple!) 5 5 2 4 4 4 4 4 4 4 4
- 60. Centrality Analysis Closeness Centrality ▸Reciprocal of the average of the shortest distances from the vertex to all other vertices. 10/19 10/18
- 61. Centrality Analysis Betweenness Centrality ▸Calculate all pairs shortest paths. ▸ Count how many times the vertex is in between the shortest paths. ▸ Useful for finding "mediators"
- 62. Centrality Analysis Application: Analysison Research Papers (2015) ▸ Joined a research project of the Institute of Statistical Mathematics in Japan. ▸ They do institutional research (IR) using governmental budgets. ▸ They wanted to develop new indices to measure researchers' productivity other than existing ones (i.e. the impact factor)
- 63. Centrality Analysis Application: Analysison Research Papers (2015)
- 64. Centrality Analysis Application: Analysison Research Papers (2015)
- 65. Centrality Analysis Application: Analysison Research Papers (2015)
- 66.
- 67. Facility Location Problems P-medianProblem ▸ Numbers represent transportation costs. ▸ We want to minimize total transportation costs with a fixed number of warehouses (p). Dr. Matt Stallmann Lightning Talk on 1/12/2018
- 68. Facility Location Problems P-medianProblem ▸ A possible solution. Total cost = (5+3+3) + (3+4+5) = 23 ▸ Can you find a better solution? Dr. Matt Stallmann Lightning Talk on 1/12/2018
- 69. Facility Location Problems P-medianProblem ▸ Complexity ▸ O(np) in general, making this NP-hard ▸ O(p2n2) for trees ▸ O(pn) for path graphs
- 70. Facility Location Problems UndergradResearch Project (Spring 2018) ▸ Developed research framework ▸ Graph generation ▸ Solving (for smaller input) ▸ Visualization ▸ Found interesting properties on trees / grids ▸ Applied linear programming to help approximation
- 71. Thank you! ▸ References ▸Wikipedia ▸ https://en.wikipedia.org/wiki/Graph_theory ▸ https://en.wikipedia.org/wiki/Eulerian_path ▸ https://en.wikipedia.org/wiki/Hamiltonian_path ▸ https://en.wikipedia.org/wiki/Graph_coloring ▸ Wolfram MathWorld ▸ http://mathworld.wolfram.com/PathGraph.html ▸ http://mathworld.wolfram.com/CycleGraph.html ▸ http://mathworld.wolfram.com/CompleteGraph.html ▸ http://mathworld.wolfram.com/BipartiteGraph.html ▸ http://mathworld.wolfram.com/GridGraph.html ▸ http://mathworld.wolfram.com/HypercubeGraph.html ▸ zyBooks: Discrete Mathematics ▸ https://www.zybooks.com/catalog/discrete-mathematics/