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Graph Theory
Farzana Yeasmin
2016-1-50-027
Fahad Alam Joy
2016-1-55-005
Maryam Mehzabin Xahin
Definition: In mathematics, graph theory is the study of graphs,
which are mathematical structures used to model pairwise
relations between objects. It refers to a set of vertices (that is
points or nodes) and of edges that connect the vertices.
Example:
Figure:1
 Now we will show how the theory of directed graphs can be used to
mathematically model relations
Directed Graphs
Definition: A directed graph is a graph that is a set of vertices connected by edges ,
where the edges have a direction associated with them.
Examples: A directed graph is a finite set of elements {P1, P2, …….. , Pn) together
with a finite collection of order pairs ( Pi , Pj) of distinct elements of this set , with no
ordered pair being repeated. The elements of the set are called vertices, and the
ordered pairs are called directed edges, of the directed graph. We use the notation
Pi → Pj( which is read, Pi connected to Pj ) belongs to the directed graph. If both Pi
→ Pj and Pj→ Pihold (denoted Pi ↔ Pj ), we draw a single line between Pi and Pj
with two oppositely pointing arrows ( as with P2 and P3 in the figure - 1 ). A directed
graph may have separate “components” of vertices that are connected only among
themselves, Like P5 in the figure – 1. Pi ↔ Pi is not permitted in a directed graph , a
vertex can not be connected with itself by a single arc that does not pass through
any other vertex.
 Represent 2 more examples of directed graphs : A directed graph having n vertices, we
may associated an n ⤬ n matrix M = [mij], called the vertex matrix of the directed
graph. Its elements are defined by
mij=
1, if Pi → Pj
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Figure: 2 Figure: 3
The corresponding vertex matrices are for figure: 2 &3
Figure: 2 Figure: 3
Graph Theory
Example 2: Figure 1 is the route map of a small airline that
services the four cities P1, P2,P3,P4. As a result graph , its vertex
matrix is
Graph Theory

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Graph Theory

  • 1. Graph Theory Farzana Yeasmin 2016-1-50-027 Fahad Alam Joy 2016-1-55-005 Maryam Mehzabin Xahin
  • 2. Definition: In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. It refers to a set of vertices (that is points or nodes) and of edges that connect the vertices. Example: Figure:1
  • 3.  Now we will show how the theory of directed graphs can be used to mathematically model relations Directed Graphs Definition: A directed graph is a graph that is a set of vertices connected by edges , where the edges have a direction associated with them. Examples: A directed graph is a finite set of elements {P1, P2, …….. , Pn) together with a finite collection of order pairs ( Pi , Pj) of distinct elements of this set , with no ordered pair being repeated. The elements of the set are called vertices, and the ordered pairs are called directed edges, of the directed graph. We use the notation Pi → Pj( which is read, Pi connected to Pj ) belongs to the directed graph. If both Pi → Pj and Pj→ Pihold (denoted Pi ↔ Pj ), we draw a single line between Pi and Pj with two oppositely pointing arrows ( as with P2 and P3 in the figure - 1 ). A directed graph may have separate “components” of vertices that are connected only among themselves, Like P5 in the figure – 1. Pi ↔ Pi is not permitted in a directed graph , a vertex can not be connected with itself by a single arc that does not pass through any other vertex.  Represent 2 more examples of directed graphs : A directed graph having n vertices, we may associated an n ⤬ n matrix M = [mij], called the vertex matrix of the directed graph. Its elements are defined by mij= 1, if Pi → Pj 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
  • 4. Figure: 2 Figure: 3 The corresponding vertex matrices are for figure: 2 &3 Figure: 2 Figure: 3
  • 6. Example 2: Figure 1 is the route map of a small airline that services the four cities P1, P2,P3,P4. As a result graph , its vertex matrix is