TreeFunctionGraph

Discrete Mathematics

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.

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.

Functions are specified as assignments as in Figure 1.

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.

Figure 2 represents a function f from A to B.

Is it a function? If not state the reasons.

One-to-Oneone-to-one or an injunction function never

assign the same value to two different domain elements.

Explain why?

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.

Explain Why?

Check these!

Bijective(both one-to-one and onto)

• If the function is both one-to-one and onto that function is bijective.

L23 15

Trees

A very important type of graph in CS is called a tree:

Real Tree transformation

L23 16

Trees

A very important type of graph in CS is called a tree:

Real Abstract

Tree Treetransformation

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

A tree with n vertices has n − 1 edges.

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.

Producing Spanning Tree

Other 4 Spanning trees of G

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.

Find Minimum Spanning Tree

We will discuss two algorithms for constructing minimum spanning trees.

Prim’s Algorithm

Solution

Try this

Kruskal’s Algorithm

Reviews from previous classes

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

Find out the diameter of the following graph

If P, A1, A2, A5, A3, A6, Q are the nodes then D=?

Isolated vertex

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

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.

Write the adjacent edges of vertex a.

An Unconnected, undirected graph

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

Write the degree of the entire graph.

L23 39

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 40

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 41

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 42

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 43

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 44

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 45

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 46

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 47

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 48

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

L23 49

Bipartite Graphs

EG: C4 is a bichromatic:

And so is bipartite, if we redraw it:

Q: For which n is Cn bipartite?

Check whether bipartite or not!

Best wishes!