electrical network graph theory topology
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Network Topology and Graph Theory
EE-304 ENT credits: 4 L3 P0 T1
Lairenlakpam Joyprakash Singh, PhD
Department of ECE,
North-Eastern Hill University (NEHU),Shillong – 793 [email protected]
August 8, 2013
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elements
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
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&
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
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N T T & D
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
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Network Toplogy Terms & Definitions
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
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Network Toplogy Terms & Definitions
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
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Network Toplogy Terms & Definitions
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
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Network Toplogy Terms & Definitions
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Network Toplogy Terms & Definitions
Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
Tree
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Network Toplogy Terms & Definitions
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Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
TreeTwigs
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Network Toplogy Terms & Definitions
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Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
TreeTwigsCo-tree
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Network Toplogy Terms & Definitions
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Introduction to Electrical Network Topology
Terms and Definitions
Circuit elementsNode
Branch
Path
Closed path or Circuit or Loop or Mesh
Topology, rather Electrical network topology
- Graph and its types
TreeTwigsCo-tree
Links or Chords
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Network Toplogy Terms & Definitions
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Network Topology: Terms and Definitions - I
Circuit elements:
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Network Toplogy Terms & Definitions
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Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,
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Network Toplogy Terms & Definitions
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Network Topology: Terms and Definitions - I
Circuit elements:
- The mathematical models of a two terminal electrical devices,- Completely characterized by its voltage-current relationship,
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Network Toplogy Terms & Definitions
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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.
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Network Toplogy Terms & Definitions
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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:
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Network Toplogy Terms & Definitions
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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 commonconnection,
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Network Toplogy Terms & Definitions
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
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Network Toplogy Terms & Definitions
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
Branch:
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Network Toplogy Terms & Definitions
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
Branch:
- A single path, containing one circuit element, which connnectsone node to any other node,
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Network Toplogy Terms & Definitions
N k l d D fi I
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
Branch:
- A single path, containing one circuit element, which connnectsone node to any other node,
- Represented by a line in the graph.
3 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
N k T l T d D fi i i I
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
Branch:
- A single path, containing one circuit element, which connnectsone node to any other node,
- Represented by a line in the graph.Path:
3 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
N t k T l T d D fi iti I
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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 commonconnection,
- The number of branches incident to a node is known as thedegree of that node.
Branch:
- A single path, containing one circuit element, which connnectsone node to any other node,
- Represented by a line in the graph.Path:
- A set of elements that may be traversed in order without passingthrough the same node twice.
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Network Toplogy Terms & Definitions
N t k T l T d D fi iti II
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Network Topology: Terms and Definitions - II
Loop:
1
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Network Toplogy Terms & Definitions
N t k T l T d D fi iti II
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Network Topology: Terms and Definitions - II
Loop:- A close path or a closed contour selected in a network/circuit,
1
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Network Toplogy Terms & Definitions
Net o k To olog Te s and Definitions II
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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 nodesthrough branches and comes to the original/starting node,
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions II
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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 nodesthrough branches and comes to the original/starting node,
- Also known as closed path or circuit .
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions II
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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 nodesthrough branches and comes to the original/starting node,
- Also known as closed path or circuit .
Mesh1 [2]:
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions II
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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 nodesthrough 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,
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions II
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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 nodesthrough 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 amesh.
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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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 nodesthrough 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 amesh.
Network:
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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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 nodesthrough 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 amesh.
Network:- The interconnection of two or more circuit elements forms an
electical network.
1
Engineering Circuit Analysis, 8e4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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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 nodesthrough 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 amesh.
Network:- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
1
Engineering Circuit Analysis, 8e4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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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 nodesthrough 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 amesh.
Network:- The interconnection of two or more circuit elements forms an
electical network.
Circuit:
- Network that contains at least one closed path,
1
Engineering Circuit Analysis, 8e4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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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 nodesthrough 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 amesh.
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.
1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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p gy
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 nodesthrough 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:
1
Engineering Circuit Analysis, 8e4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - II
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p gy
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 nodesthrough 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.1
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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p gy
Topology:
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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p gy
Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.
5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.
Graph:
5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.
Graph:- A graph corresponding to a given network is obtained byreplacing all circuit elements with lines.
5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.
Graph:- A graph corresponding to a given network is obtained byreplacing all circuit elements with lines.
- Connected graph: A graph in which at least one path existsbetween any two nodes of the graph. If the network has atransformer as one of the element, then the resulted graph isunconnected
5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare arranged.
Graph:- A graph corresponding to a given network is obtained byreplacing all circuit elements with lines.
- Connected graph: A graph in which at least one path existsbetween any two nodes of the graph. If the network has atransformer as one of the element, then the resulted graph isunconnected
- Directed or Oriented graph: A graph that has all the nodesand branches numbered and also directions are given to thebranches.
5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - III
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Topology:
- Deals with properties of networks which are unaffected when the
network is stretched, twisted, or otherwise distorted the size andthe shape,
- Not concerned with the particular types of elements appearing inthe circuit, but only with the way in which branches and nodesare 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 existsbetween any two nodes of the graph. If the network has atransformer as one of the element, then the resulted graph isunconnected
- Directed or Oriented graph: A graph that has all the nodesand branches numbered and also directions are given to thebranches.
- Subgraph: The subset of a graph. If the number of nodes andbranches of a subgraph is less than that of the graph, thesubgraph is said to be proper.
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Network Toplogy Network Circuits & Their Graphs
Network Topology: An example
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A circuit with topologically equivalent graphs:
12
3
4
+
−V s
I s
R1I R1
R2I R2
R3
I R3
C
I C
LI L
12
3
4
12
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).
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Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - I
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1 2 3
4
5 A
R1 R2
R4C R3
(a)
12
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 equalto the number of nodes or vertices.
There are at least two branches in a circuit.
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Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - II
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1 2 3
4
5 A
R1I R1R2I R2
R4
I R4
C
I C
R3
I R3
(a)
12
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 anarrow, and the resulted graph is called oriented/directed graph .
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Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - III
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12
3
4
5
+
−V s
I s
R
R1I 1
R2I 2
R3
I 3
C
I C
LI L
(a)
12
3
4
5
a
b c
f e
d
g
(b)
12
3
4, 5
a
b c
f ed
(c)
Figure 3 : (a) A circuit, (b) its directed graph and (c) simplified directedgraph of (b).
Note:
The active element branch is replaced by its internal resistance tosimplify analysis and computation.
9 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - IV
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12
3
4
+
−V s
I vs
R
R1I 1
R2I 2
I I sC
I C
L
I L
(a)
12
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 analysisand computation.
1 0 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Network Circuits & Their Graphs
An Electrical Network & its Graph - V
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1A
1Ω
1Ω
1Ω
1Ω
1Ω
+ −1V
(a) (b) (c)
Figure 5 : (a) A circuit, and its- (b) simplified graph and (c) directedgraph.
Note:
When voltage source is not in series with any passive element in thegiven network, it is kept in the graph as a branch.
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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Tree:
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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Tree:
- A connected subgraph having all the nodes of a graph without
any loop.
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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.
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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:
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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 ,
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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:
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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 atree are termed as links or chords ,
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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 atree are termed as links or chords ,
- Links are complement of twigs.
1 2 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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 towhich 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 atree are termed as links or chords ,
- Links are complement of twigs.
Co-tree:
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Network Toplogy Terms & Definitions
Network Topology: Terms and Definitions - IV
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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 towhich 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 atree are termed as links or chords ,
- Links are complement of twigs.
Co-tree:
- The graph constituted with links is known as c o-tr ee .
1 2 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Tree and Cotree
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Given a Graph:1 2 3
4
a b
c d ef
Tree Twigs of tree Links of cotree
12 3
4
a b
c d ef
a,b,d c,e,f
12 3
4
a b
c d ef
a,d,f c,b,e
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Network Toplogy Terms & Definitions
Summary and a Question:
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Q. Does the following graph with branches a and e form a tree?
12
3
4
a
b c
f ed
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Network Toplogy Terms & Definitions
Summary and a Question:
Q D h f ll i h i h b h d f ?
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Q. Does the following graph with branches a and e form a tree?
12
3
4
a
b c
f ed
The number of nodes in this subgraph isequal to that of the given graph.
1 4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy Terms & Definitions
Summary and a Question:
Q D th f ll i h ith b h d f t ?
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Q. Does the following graph with branches a and e form a tree?
12
3
4
a
b c
f ed
The number of nodes in this subgraph isequal to that of the given graph.
But it has unconnected subgraphs andmoreover total number branches
= n− 1(= 3). Therefore, it is not a tree.
1 4 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy References
Text Books & References
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M. E. Van ValkenburgNetwork Analysis, 3/e .
PHI, 2005.
W.H. Hayt, J.E. Kemmerly, S.M. DurbinEngineering Circuit Analysis, 8/e .MH, 2012.
1 5 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Network Toplogy References
Text Books & References
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M. E. Van ValkenburgNetwork Analysis, 3/e .
PHI, 2005.
W.H. Hayt, J.E. Kemmerly, S.M. DurbinEngineering Circuit Analysis, 8/e .MH, 2012.
M. Nahvi, J.A. EdministerSchuamâĂŹs Outline Electric Circuits, 4/e .TMH, SIE, 2007.
A. Sudhakar, S.S. Palli
Circuits and Networks: Analysis and Synthesis, 2/e .TMH, 2002.
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Network Toplogy Khublei Shibun!
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Thank You!
Any Question?
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Home Assignment Graph and Incidence Matrix
Problems for Practice: Graph and Incidence Matrix
1 Classify whether each of the following graphs as planar or
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1. Classify whether each of the following graphs as planar ornonplanar .
2. Find the number of possible trees for each graph and draw allpossible trees.
1
2
3
4
(a)
ab c
d
(b)
1 2
3
4
5
(c)
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Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: While replacing all elements of the network with lines to form a
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Note: While replacing all elements of the network with lines to form agraph, we replace active elements by their internal resistances
to simplify analysis and computation.For example - 1:
1
23 4
5
+
−V s
R1
I s
R2I 1
R3I 2
R4
I 3
C
I C
LI L
I s
(a)
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Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: While replacing all elements of the network with lines to form a
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Note: While replacing all elements of the network with lines to form agraph, we replace active elements by their internal resistances
to simplify analysis and computation.For example - 1:
1
23 4
5
+
−V s
R1
I s
R2I 1
R3I 2
R4
I 3
C
I C
LI L
I s
(a)
23
4
1, 5
(b)
1 8 / 2 0 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Network topology and graph
Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: Transformer gives a unconnected graph!
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Note: Transformer gives a unconnected graph!For example - 2:
12 3 4
56
+
−V s
R1
I 1
R2I 2
K
C I C
R3
I 3
I
(a)
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Home Assignment Graph and Incidence Matrix
Problems for Practice - II
Note: Transformer gives a unconnected graph!
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Note: Transformer gives a unconnected graph!For example - 2:
12 3 4
56
+
−V s
R1
I 1
R2I 2
K
C I C
R3
I 3
I
(a)
12
6
34
5
(b)
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Home Assignment Graph and Incidence Matrix
Few non-planar Graphs
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1
2
3
4
(a)
1 2
3
45
6
(b)
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