L6FM1 Further Maths DiscreteDecision Maths Dr Cooper NSC - PowerPoint PPT Presentation

Title: L6FM1 Further Maths DiscreteDecision Maths Dr Cooper NSC


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L6FM1Further MathsDiscrete/Decision MathsDr
Cooper (NSC)
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Graph Theory
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Graph Theory
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Graph Theory
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Biochemical Networks
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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)
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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
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Graph Theory
Simple graphs do not have loops or multiple arcs
between pairs of nodes. Most networks in D1 are
Simple graphs.
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Graph Theory
Simple graphs do not have loops or multiple arcs
between pairs of nodes. Most networks in D1 are
Simple graphs.
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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
K4
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Graph Theory
A subgraph can be formed by removing arcs and/or
nodes from another graph.
Subgraph
Graph
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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.
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Graph Theory
A complete bipartite graph is a bipartite graph
in which
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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
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Graph Theory
B
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.
C
A
D
F
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Graph Theory
B
A connected graph is one for which a path can be
found between any two nodes. The illustrated
graph is NOT connected.
C
A
D
X
F
Y
Z
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Graph Theory
E
B
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
C
A
D
F
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Graph Theory
B
A semi-Eulerian Graph has exactly two nodes of
odd order. Such graphs contain a non-closed
trail containing every arc precisely once.
C
A
D
F
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Graph Theory
B
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
C
A
D
F
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Konigsberg Bridges
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Konigsberg Bridges