A gentle introduction of Graph Theoretic Concepts

1. L6FM1
Further Maths
Discrete/Decision Maths
Dr Cooper (NSC)

2. Graph Theory

3. Graph Theory

4. Graph Theory

6. Biochemical Networks

8. 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)

9. 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

10. Graph Theory
Simple graphs do not have
loops or multiple arcs between
pairs of nodes. Most networks in
D1 are Simple graphs.

11. Graph Theory
Simple graphs do not have
loops or multiple arcs between
pairs of nodes. Most networks in
D1 are Simple graphs.

12. Graph Theory
A complete graphs is one in
which every node is connected K4
to every other node. The notation
for the complete graph with n
nods is Kn

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

14. 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.

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

16. 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

17. Graph Theory
B
The order of a node is the number C
of arcs meeting at that node. A
D
In the subgraph shown, A and F
have order 2, B and C have order 3 F
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.

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

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

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

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

22. Konigsberg Bridges

23. Konigsberg Bridges