Electrical Network Topology, Electrical Network Graph Theory, Node, Branch, Twig, Link, Tree, Cotree

1. Network Topology and Graph Theory
EE-304 ENT credits: 4 L{3} P{0} T{1}
Lairenlakpam Joyprakash Singh, PhD
Department of ECE,
North-Eastern Hill University (NEHU),
Shillong – 793 022
jplairen@nehu.ac.in
August 8, 2013
1 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

2. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

3. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

4. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

5. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

6. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

7. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

8. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

9. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

10. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
Tree
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

11. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
Tree
Twigs
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

12. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
Tree
Twigs
Co-tree
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

13. Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
Node
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
Tree
Twigs
Co-tree
Links or Chords
2 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

14. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

15. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

16. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

17. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

18. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

19. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

20. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

21. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
Branch:
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

22. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
Branch:
- A single path, containing one circuit element, which connnects
one node to any other node,
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

23. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
Branch:
- A single path, containing one circuit element, which connnects
one node to any other node,
- Represented by a line in the graph.
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

24. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
Branch:
- A single path, containing one circuit element, which connnects
one node to any other node,
- Represented by a line in the graph.
Path:
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

25. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
- Completely characterized by its voltage-current relationship,
- Can not be subdivided into other two-terminal devices.
Node:
- A point at which two or more circuit elements have a common
connection,
- The number of branches incident to a node is known as the
degree of that node.
Branch:
- A single path, containing one circuit element, which connnects
one node to any other node,
- Represented by a line in the graph.
Path:
- A set of elements that may be traversed in order without passing
through the same node twice.
3 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

26. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

27. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

28. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

29. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

30. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

31. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

32. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

33. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

34. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

35. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

36. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
- Network that contains at least one closed path,
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

37. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
- Network that contains at least one closed path,
- Every circuit is a network, but not all networks are circuits.
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

38. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
- Network that contains at least one closed path,
- Every circuit is a network, but not all networks are circuits.
Planar circuit:
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

39. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
Loop:
- A close path or a closed contour selected in a network/circuit,
- A path that may be started from a prticular node to other nodes
through branches and comes to the original/starting node,
- Also known as closed path or circuit.
Mesh1
[2]:
- A loop that does not contain any other loops within it,
- Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
Network:
- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
- Network that contains at least one closed path,
- Every circuit is a network, but not all networks are circuits.
Planar circuit:
- A circuit that may drawn on a plane surface in such a way that
no branch passes over or under any other branch.
1Engineering Circuit Analysis, 8e
4 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

40. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

41. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

42. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

43. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

44. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
- A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

45. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
- A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
- Connected graph: A graph in which at least one path exists
between any two nodes of the graph. If the network has a
transformer as one of the element, then the resulted graph is
unconnected
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

46. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
- A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
- Connected graph: A graph in which at least one path exists
between any two nodes of the graph. If the network has a
transformer as one of the element, then the resulted graph is
unconnected
- Directed or Oriented graph: A graph that has all the nodes
and branches numbered and also directions are given to the
branches.
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

47. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size and
the shape,
- Not concerned with the particular types of elements appearing in
the circuit, but only with the way in which branches and nodes
are arranged.
Graph:
- A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
- Connected graph: A graph in which at least one path exists
between any two nodes of the graph. If the network has a
transformer as one of the element, then the resulted graph is
unconnected
- Directed or Oriented graph: A graph that has all the nodes
and branches numbered and also directions are given to the
branches.
- Subgraph: The subset of a graph. If the number of nodes and
branches of a subgraph is less than that of the graph, the
subgraph is said to be proper.
5 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

48. Network Toplogy Network Circuits & Their Graphs
Network Topology: An example
A circuit with topologically equivalent graphs:
1
2
3
4
+
−Vs
Is
R1
IR1
R2
IR2
R3
IR3
C
IC
L
IL
1
2
3
4
1
2
3
4
i) A Circuit ii) its graph iii) directed graph
1
2
3
4
1
2
3
4
1
2
3
4
Three topologically equivalent graphs of figure ii).
6 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

49. Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - I
1 2 3
4
5 A
R1 R2
R4CR3
(a)
1
2
3
4
a b
c
d e f
(b)
Figure 1 : (a) A circuit and (b) its graph.
Note:
The maximum number of branches possible, in a circuit, will be equal
to the number of nodes or vertices.
There are at least two branches in a circuit.
7 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

50. Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - II
1 2 3
4
5 A
R1
IR1
R2
IR2
R4
IR4
C
IC
R3
IR3
(a)
1
2
3
4
a b
c
d e f
(b)
Figure 2 : (a) A circuit and (b) its directed graph.
Note:
Each of the lines of the graph is indicated a reference direction by an
arrow, and the resulted graph is called oriented/directed graph.
8 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

51. Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - III
1
2
3
4
5
+
−Vs
Is
R
R1
I1
R2
I2
R3
I3
C
IC
L
IL
(a)
1
2
3
4
5
a
b c
fe
d
g
(b)
1
2
3
4, 5
a
b c
fed
(c)
Figure 3 : (a) A circuit, (b) its directed graph and (c) simplified directed
graph of (b).
Note:
The active element branch is replaced by its internal resistance to
simplify analysis and computation.
9 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

52. Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - IV
1
2
3
4
+
−Vs
Ivs
R
R1
I1
R2
I2
IIsC
IC
L
IL
(a)
1
2
3
4
a
b c
ed
(b)
1
2
3
4
a
b c
e
d
(c)
Figure 4 : (a) A circuit, and (b),(c) its directed graphs.
Note:
The active elements are excluded from the graph to simplify analysis
and computation.
10 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

53. Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - V
1 A
1 Ω
1 Ω
1 Ω
1 Ω
1 Ω
+−
1 V
(a) (b) (c)
Figure 5 : (a) A circuit, and its- (b) simplified graph and (c) directed
graph.
Note:
When voltage source is not in series with any passive element in the
given network, it is kept in the graph as a branch.
11 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

54. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

55. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

56. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

57. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

58. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

59. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords:
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

60. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords:
- The branches that are removed from the graph while forming a
tree are termed as links or chords,
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

61. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords:
- The branches that are removed from the graph while forming a
tree are termed as links or chords,
- Links are complement of twigs.
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

62. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords:
- The branches that are removed from the graph while forming a
tree are termed as links or chords,
- Links are complement of twigs.
Co-tree:
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

63. Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
- Thus, a tree is a subgraph that has the following properties:
- It must consist of all nodes of a complete graph.
- For a graph having n number of nodes, the tree of the given graph
will have n − 1 branches.
- There exists one and only one path between any pair of nodes.
- A tree should not have any closed path.
- The rank of a tree is (n − 1). This is also the rank of the graph to
which the tree belongs.
Twigs:
- The branches of a tree are known as twigs,
Links or Chords:
- The branches that are removed from the graph while forming a
tree are termed as links or chords,
- Links are complement of twigs.
Co-tree:
- The graph constituted with links is known as co-tree.
12 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

64. Network Toplogy Terms & Definitions
Tree and Cotree
Given a Graph:
1 2 3
4
a b
c d e
f
Tree Twigs of tree Links of cotree
1
2 3
4
a b
c d e
f
{a,b,d} {c,e,f}
1
2 3
4
a b
c d e
f
{a,d,f} {c,b,e}
13 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

65. Network Toplogy Terms & Definitions
Summary and a Question:
Q. Does the following graph with branches a and e form a tree?
1
2
3
4
a
b c
fed
14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

66. Network Toplogy Terms & Definitions
Summary and a Question:
Q. Does the following graph with branches a and e form a tree?
1
2
3
4
a
b c
fed

67. The number of nodes in this subgraph is
equal to that of the given graph.
14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

68. Network Toplogy Terms & Definitions
Summary and a Question:
Q. Does the following graph with branches a and e form a tree?
1
2
3
4
a
b c
fed

69. The number of nodes in this subgraph is
equal to that of the given graph.

70. But it has unconnected subgraphs and
moreover total number branches
= n − 1(= 3). Therefore, it is not a tree.
14 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

71. Network Toplogy References
Text Books & References
M. E. Van Valkenburg
Network Analysis, 3/e.
PHI, 2005.
W.H. Hayt, J.E. Kemmerly, S.M. Durbin
Engineering Circuit Analysis, 8/e.
MH, 2012.
15 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

72. Network Toplogy References
Text Books & References
M. E. Van Valkenburg
Network Analysis, 3/e.
PHI, 2005.
W.H. Hayt, J.E. Kemmerly, S.M. Durbin
Engineering Circuit Analysis, 8/e.
MH, 2012.
M. Nahvi, J.A. Edminister
SchuamâĂŹs Outline Electric Circuits, 4/e.
TMH, SIE, 2007.
A. Sudhakar, S.S. Palli
Circuits and Networks: Analysis and Synthesis, 2/e.
TMH, 2002.
15 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

73. Network Toplogy Khublei Shibun!
Thank You!
Any Question?
16 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

74. Home Assignment Graph and Incidence Matrix
Problems for Practice: Graph and Incidence Matrix
1. Classify whether each of the following graphs as planar or
nonplanar.
2. Find the number of possible trees for each graph and draw all
possible trees.
1
2
3
4
(a)
a
b c
d
(b)
1 2
3
4
5
(c)
17 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

75. Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: While replacing all elements of the network with lines to form a
graph, we replace active elements by their internal resistances
to simplify analysis and computation.
For example - 1:
1
2
3 4
5
+
−Vs
R1
Is
R2
I1
R3
I2
R4
I3
C
IC
L
IL
Is
(a)
18 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

76. Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: While replacing all elements of the network with lines to form a
graph, we replace active elements by their internal resistances
to simplify analysis and computation.
For example - 1:
1
2
3 4
5
+
−Vs
R1
Is
R2
I1
R3
I2
R4
I3
C
IC
L
IL
Is
(a)
2
3
4
1, 5
(b)
18 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

77. Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: Transformer gives a unconnected graph!
For example - 2:
1
2 3 4
56
+
−Vs
R1
I1
R2
I2
K
C
IC
R3
I3
I
(a)
19 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

78. Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: Transformer gives a unconnected graph!
For example - 2:
1
2 3 4
56
+
−Vs
R1
I1
R2
I2
K
C
IC
R3
I3
I
(a)
1
2
6
3
4
5
(b)
19 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph

79. Home Assignment Graph and Incidence Matrix
Few non-planar Graphs
1
2
3
4
(a)
1 2
3
45
6
(b)
20 / 20 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph