L 6 FM 1 Further Maths DiscreteDecision Maths

L 6 FM 1 Further Maths Discrete/Decision Maths Dr Cooper (NSC)

Graph Theory

Graph Theory

Graph Theory

Biochemical Networks

Graph Theory A graph (network) is a collection of nodes (also called vertices, shown by blobs) connected by arcs (or edges or legs, shown by straight or curved lines)

Graph Theory A graph (network) is a collection of nodes (also called vertices, shown by blobs) connected by arcs (or edges or legs, shown by 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

Graph Theory Simple graphs do not have loops or multiple arcs between pairs of nodes. Most networks in D 1 are Simple graphs.

Graph Theory Simple graphs do not have loops or multiple arcs between pairs of nodes. Most networks in D 1 are Simple graphs.

Graph Theory A complete graphs is one in which every node is connected to every other node. The notation for the complete graph with n nods is Kn K 4

Graph Theory A subgraph can be formed by removing arcs and/or nodes from another graph. Graph Subgraph

Graph Theory A bipartite graph is a graph in which there are 2 sets of nodes. There are no arcs within either set of nodes.

Graph Theory A complete bipartite graph is a bipartite graph in which …

Graph Theory A complete bipartite graph is a bipartite graph in which every node in one set is connected to every node in the other set

Graph Theory The order of a node is the number of arcs meeting at that node. In the subgraph shown, A and F have order 2, B and C have order 3 and D has order 4. A, D and F have even order, B and C odd order. Since every arc adds 2 to the total order of all the nodes, this total is always even. B A C D F

Graph Theory B A connected graph is one for which a path can be found between A any two nodes. C D X The illustrated graph is NOT connected. Z Y F

Graph Theory An Eulerian Graph has every node of even order. Euler proved that this was identical to there being a closed trail containing every arc precisely once. e. g. BECFDABCDB E B A C D F

Graph Theory A semi-Eulerian Graph has exactly two nodes of odd order. Such graphs contain a non-closed trail containing every arc precisely once. B A C D F

Graph Theory A semi-Eulerian Graph has exactly two nodes of odd order. Such graphs contain a non-closed trail containing every arc precisely once. Such a trail must start at one odd node and finish at the other. e. g. BADBCDFC B A C D F

Konigsberg Bridges

Konigsberg Bridges