1. GRAPH Learning Outcomes Students should be able to:
Explain basic terminology of a graph
Identify Euler and Hamiltonian cycle
Represent graphs using adjacency matrices

2. Contents Introduction Paths and circuits
Matrix representations of graphs

3. Introduction to Graphs
DEF: A simple graph G = (V,E ) consists of a non-empty set V of vertices (or nodes) and a set E (possibly empty) of edges where each edge is associated with a set consisting of either one or two vertices called its endpoints.
Q: For a set V with n elements, how many possible edges there?

4. Terminology Loop, parallel edges, isolated, adjacent
Loop - an edge connects a vertex to itself
Two edges connect the same pair of vertices are said to be parallel.
Isolated vertex – unconnected vertex.
Two vertices that are connected by an edge are called adjacent.
An edge is said to be incident on each of its end points.

5. Example of a graph Vertex set = {u1, u2, u3}
Edge set = {e1, e2, e3, e4}
e1, e2, e3 are incident on u1
u2 and u3 are adjacent to u1
e4 is a loop
e2 and e3 are parallel

6. Types of Graphs Directed – order counts when discussing edges
Undirected (bidirectional)
Weighted – each edge has a value associated with it
Unweighted
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8. Special Graphs Simple – does not have any loops or parallel edges
Complete graphs – there is an edge “between” every possible tuple of vertices
Bipartite graph – V can be partitioned into V1 and V2, such that:
(x,y)E  (xV1  yV2)  (xV2  yV1)
Sub graphs
G1 is a subset of G2 iff
Every vertex in G1 is in G2
Every edge in G1 is in G2
Connected graph – can get from any vertex to another via edges in the graph
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9. Complete Graphs
there is an edge “between” every possible tuple of vertices. |e| = C(n,2) = n. (n-1)/2

10. Bipartite graph
A graph is bipartite if its vertices can be partitioned into two disjoint subsets U and V such that each edge connects a vertex from U to one from V.

11. Complete bipartite
A bipartite graph is a complete bipartite graph if every vertex in U is connected to every vertex in V. If U has m elements and V has n, then we denote the resulting complete bipartite graph by Km,n. The illustration shows K3,2

12. Degree of Vertex
Defined as the number of edges attached (incident) to the vertex. A loop is counted twice.
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13. Handshake Theorem
If G is any graph, then the sum of the degrees of all the vertices of G equals twice the number of edges of G.
Specifically, if the vertices of G are v1, v2, …, vn, where n is a nonnegative integer, then:
The total degree of G = d(v1)+d(v2)+…+d(vn)
= 2  (the number of edges of G)
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14. Total degree of a graph is even
Prove that the total of the degrees of all vertices in a graph is even.
Since the total degree equals 2 times of edges, which is an integer, the sum of all degree is even.
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15. Whether certain graphs exist
Draw a graph with the specified properties or show that no such graph exists.
(a) A graph with four vertices of degrees 1,1,2, and 3
(b) A graph with four vertices of degrees 1,1,3 and 3
(c) A simple graph with four vertices of degrees 1,1,3 and 3

16. Even no. of vertices with odd degree
In any graph, there are an even number of vertices with odd degree
Is there a graph with ten vertices of degrees 1,1,2,2,2,3,4,4,4, and 6?
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17. Learning Outcomes Students should be able to:
Explain basic terminology of a graph
Identify Euler and Hamiltonian cycle
Represent graphs using adjacency matrices

18. Seven Bridges of Königsberg
Is it possible for a person to take a walk around town, starting and ending at the same location and crossing each of the seven bridges exactly once? No
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19. Definitions
Terminology - Walk, path, simple path, circuit, simple circuit.
Walk from two vertices is a finite alternating sequence of adjacent vertices and edges
Trivial walk from v to v consists of single vertex
v0e1v1e2…envn
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20. Path
Path – a walk that does not contain a repeated edge (may have a repeated vertex)
v0e1v1e2…envn where all the ei are distinct
Simple path – a path that does not contain a repeated vertex
v0e1v1e2…envn where all the ei and vj are distinct

21. Circuit Closed walk – starts and ends at same vertex
Circuit – a closed walk without repeated edge
Simple circuit – a circuit with no repeated vertex except first and last

22. Connectedness
Connectedness – if there is a walk from one to the other

23. Euler Circuits
A circuit that contains every vertex and every edge of G.
A sequence of adjacent vertices and edges
that starts and ends at the same vertex,
uses every vertex of G at least once, and
uses every edge of G exactly once.
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24. If a graph has an Euler circuit, every vertex has even degree.
Contrapositive: if some vertex has odd degree, then the graph does not have an Euler circuit.
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25. If every vertex of nonempty graph has even degree and if graph is connected, then the graph has an Euler circuit.
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26. Euler Circuit
A graph G has an Euler circuit if, and only if, G is connected and every vertex of G has even degree.
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27. Hamiltonian Path
A path in an undirected graph which visits each vertex exactly once.
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28. Hamiltonian Circuit A simple circuit that includes every vertex of G.
A sequence of adjacent vertices and distinct edges in which every vertex of G appears exactly once, except for the first and last, which are the same.
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29. Hamiltonian Circuit
Proved simple criterion for determining whether a graph has an Euler circuit
No analogous criterion for determining whether a graph has a Hamiltonian circuit
Nor is there an efficient algorithm for finding such an algorithm
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30. Traveling Salesman Problem
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31. TSP One way to solve the general problem is to:
Write down all Hamiltonian circuits
Compute total distance for each
Pick one for which total is minimal
What if graph has 30 vertices:
29! =8.84 x 1030 different Hamiltonian circuits
If each circuit could be found and total distance computed in a nanosecond, then would take:
2.8 x 1014 years!!!
No known algorithm that is more efficient!!!
Some that find “pretty good” solutions
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32. Learning Outcomes Students should be able to:
Explain basic terminology of a graph
Identify Euler and Hamiltonian cycle
Represent graphs using adjacency matrices

33. Matrix Representations of Graphs
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34. Matrices and Connected Components
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35. Counting Walks of Length n
Matrix multiplication
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36. How do these graphs relate?=   
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37. Summary