Graphs

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A gentle introduction of Graph Theoretic Concepts

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Graphs

  1. 1. L6FM1 Further MathsDiscrete/Decision Maths Dr Cooper (NSC)
  2. 2. Graph Theory
  3. 3. Graph Theory
  4. 4. Graph Theory
  5. 5. Biochemical Networks
  6. 6. Graph TheoryA graph (network) is a collectionof nodes (also called vertices,shown by blobs) connected byarcs (or edges or legs, shownby straight or curved lines)
  7. 7. Graph TheoryA graph (network) is a collectionof nodes (also called vertices,shown by blobs) connected byarcs (or edges or legs, shownby straight or curved lines) Graphs can used to represent oil flow in pipes, traffic flow on motorways, transport of pollution by rivers, groundwater movement of contamination, biochemical pathways, the underground network, etc
  8. 8. Graph TheorySimple graphs do not haveloops or multiple arcs betweenpairs of nodes. Most networks inD1 are Simple graphs.
  9. 9. Graph TheorySimple graphs do not haveloops or multiple arcs betweenpairs of nodes. Most networks inD1 are Simple graphs.
  10. 10. Graph TheoryA complete graphs is one inwhich every node is connected K4to every other node. The notationfor the complete graph with nnods is Kn
  11. 11. Graph TheoryA subgraph can be formed by removing arcs and/or nodesfrom another graph. Graph Subgraph
  12. 12. Graph TheoryA bipartite graph is a graph in which there are 2 sets ofnodes. There are no arcs within either set of nodes.
  13. 13. Graph TheoryA complete bipartite graph is a bipartite graph in which …
  14. 14. Graph TheoryA complete bipartite graph is a bipartite graph in whichevery node in one set is connected to every node in the otherset
  15. 15. Graph Theory BThe order of a node is the number Cof arcs meeting at that node. A DIn the subgraph shown, A and Fhave order 2, B and C have order 3 Fand D has order 4. A, D and F haveeven order, B and C odd order.Since every arc adds 2 to the totalorder of all the nodes, this total isalways even.
  16. 16. Graph Theory BA connected graph is one for Cwhich a path can be found between Aany two nodes. D XThe illustrated graph is NOT Y Fconnected. Z
  17. 17. Graph Theory E BAn Eulerian Graph has every Cnode of even order. A DEuler proved that this was identicalto there being a closed trail Fcontaining every arc preciselyonce. e.g. BECFDABCDB
  18. 18. Graph Theory BA semi-Eulerian Graph has Cexactly two nodes of odd order. A DSuch graphs contain a non-closedtrail containing every arc precisely Fonce.
  19. 19. Graph Theory BA semi-Eulerian Graph has Cexactly two nodes of odd order. A DSuch graphs contain a non-closedtrail containing every arc precisely Fonce.Such a trail must start at one oddnode and finish at the other.e.g. BADBCDFC
  20. 20. Konigsberg Bridges
  21. 21. Konigsberg Bridges

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