S GraphS GraphS Graphs Specification Language Jose Domingo López López Ángel Escribano Santamarina.
Graphs
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Transcript of Graphs
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L6FM1
Further Maths
Discrete/Decision Maths
Dr Cooper (NSC)
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Graph Theory
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Graph Theory
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Graph Theory
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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
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
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
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.
Graph Subgraph
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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
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.
A
BC
D
F
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Graph Theory
A connected graph is one for which a path can be found between any two nodes.
The illustrated graph is NOT connected.
A
BC
D
FX
Y
Z
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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
A
BC
D
F
E
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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.
A
BC
D
F
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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
A
BC
D
F
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Konigsberg Bridges
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Konigsberg Bridges