For graphs of mathematical functions, see Graph of a function. For other uses, see Graph (disambiguation). A drawing of a graph. In mathematics graph theory is the study of graphs, which are mathematical structures used.In mathematics, and more specifically in graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path. In other words, any acyclic connected graph is a tree. A forest is a disjoint union of trees.

1. Tree
Function
Graph
Discrete Mathematics

2. Function Requirements
There are rules for functions to be well defined, or correct.
• No element of the domain must be left unmapped.
• No element of the domain may map to more than one
element of the co-domain.

4. Definition 1
• Let A and B be nonempty sets.
A function f from A to B is an
assignment of exactly one
element of B to each element
of A.

5. Functions are specified as assignments as in Figure 1.

6. DEFINITION 2
If f is a function from A to
B, we say that A is the
domain of f and B is the
co-domain of f.

7. Figure 2 represents a function f from A to B.

8. Is it a function? If not state
the reasons.

9. One-to-One
one-to-one or an injunction function never
assign the same value to two different domain
elements.

10. Explain why?

11. When every member of the codomain is the image of some element of
the domain then functions are called onto functions or a surjection.
Onto Functions.

12. Explain Why?

13. Check these!

14. Bijective(both one-to-one and
onto)
• If the function is both one-to-
one and onto that function is
bijective.

15. L23 15
Trees
A very important type of graph in CS is called a
tree:
Real
Tree
transformation

16. L23 16
Trees
A very important type of graph in CS is called a
tree:
Real Abstract
Tree Tree
transformation

17. Tree!A tree is a connected undirected graph with no simple circuits.
Find which of them are not trees and why!

18. A tree with n vertices has n − 1 edges.

20. Spanning Trees
• Let G be a simple graph. A spanning
tree of G is a subgraph of G that is a
tree containing every vertex of G.
• A simple graph with a spanning tree
must be connected, because there is
a path in the spanning tree between
any two vertices.

21. Producing Spanning Tree

22. Other 4 Spanning trees of G

23. Minimum Spanning Trees
A minimum spanning tree in a
connected weighted graph is a
spanning tree that has the
smallest possible sum of weights
of its edges.

24. Find Minimum Spanning Tree

25. We will discuss two
algorithms for
constructing minimum
spanning trees.

26. Prim’s Algorithm

27. Solution

28. Try this

29. Kruskal’s Algorithm

31. Reviews from previous
classes

32. The diameter of a graph is the length of the shortest path between the most distanced
nodes. The highest value of the topological distance of this matrix is the diameter of
the graph (d=4).
Diameter of a Graph

33. Find out the diameter of the following
graph
If P, A1, A2, A5, A3, A6, Q are the nodes then D=?

34. Isolated vertex
• A vertex of degree zero is called isolated. It
follows that an isolated vertex is not adjacent
to any vertex.

35. Adjacent edge
• When two vertices of a graph are
connected by an edge, these
vertices are adjacent vertices.
• When two edges meet at the
same vertex, those two edges are
said to be adjacent.

36. Write the adjacent edges of vertex a.

37. An Unconnected, undirected graph
• Clearly, the graph is not connected.However, the figure actually represents
the undirected graph G8=(v , e) , given by

38. Write the degree of the entire graph.

39. L23 39
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

40. L23 40
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

41. L23 41
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

42. L23 42
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

43. L23 43
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

44. L23 44
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

45. L23 45
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

46. L23 46
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

47. L23 47
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

48. L23 48
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:

49. L23 49
Bipartite Graphs
EG: C4 is a bichromatic:
And so is bipartite, if we redraw it:
Q: For which n is Cn bipartite?

50. Check whether bipartite or not!

51. Best wishes!