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
[email protected]
August 8, 2013
1 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Network Toplogy
Terms & Definitions
Network Topology: Terms and Definitions - II Loop:
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
Network Toplogy
Terms & Definitions
Network Topology: Terms and Definitions - II Loop: - A close path or a closed contour selected in a network/circuit,
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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,
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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.
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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]:
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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,
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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.
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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:
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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.
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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:
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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,
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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.
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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:
1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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. 1 Engineering 4 / 20
Circuit Analysis, 8e
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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
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
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
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
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
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
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
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
Network Toplogy
Network Circuits & Their Graphs
Network Topology: An example A circuit with topologically equivalent graphs: L
IL IR1
1
R1
2 I R2
Is
2
1
3
3
2
1
3
IR3
IC − +
Vs
R2
R3
C
4
4
i) A Circuit
4
ii) its graph
iii) directed graph
2 2
2 1
3 1
3
4 4
4 1
3
Three topologically equivalent graphs of figure ii). 6 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
Network Toplogy
Network Circuits & Their Graphs
An Electrical Network & its Graph - I
R1
1
R3
5A
R2
2
C
3
a
1
R4
b
2
e
d
3
f
c
4
4
(a)
(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
Network Toplogy
Network Circuits & Their Graphs
An Electrical Network & its Graph - II
1 IR1
R1
IR3
R2
3
a
1
b
2
3
IR4
IC
R3
5A
2 IR2
C
R4
e
d
f
c
4
4
(a)
(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
Network Toplogy
Network Circuits & Their Graphs
An Electrical Network & its Graph - III
IL
L a
a I1
1
R1
2 I2
R2 3
b
1
c
2
3
b
1
c
2
3
I3
IC
R
d 5
C
Is
R3
e 5
− +
Vs
f
d
e
f
g 4
(a)
4
(b)
4, 5
(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
Network Toplogy
Network Circuits & Their Graphs
An Electrical Network & its Graph - IV
L
IL
a I1
1
R1
2 I2
R2 3
b
1
3
a
1
3 c
b C
Ivs
Is d
e
d
2
− +
Vs
c
2
IC
R
e 4
(a)
4
4
(b)
(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
Network Toplogy
Network Circuits & Their Graphs
An Electrical Network & its Graph - V
1A
1V
−+
1Ω 1Ω
1Ω 1Ω
1Ω
(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
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
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
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
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
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
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
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
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
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
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
Network Toplogy
Terms & Definitions
Tree and Cotree Given a Graph: 1 a 2 b 3 d e
c
f
4
Tree 1
d e
c
Links of cotree
{a,b,d}
{c,e,f}
{a,d,f}
{c,b,e}
f
4 a 2 b 3
1
d e
c 4
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Twigs of tree
a 2 b 3
f
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
Network Toplogy
Terms & Definitions
Summary and a Question: Q. Does the following graph with branches a and e form a tree? a
b
1
d
2
c
e
3
f
4
14 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
Network Toplogy
Terms & Definitions
Summary and a Question: Q. Does the following graph with branches a and e form a tree? a
b
1
d
2
c
e
3
+ The number of nodes in this subgraph is equal to that of the given graph.
f
4
14 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
Network Toplogy
Terms & Definitions
Summary and a Question: Q. Does the following graph with branches a and e form a tree? a
b
1
d
2
c
e
3
f
+ The number of nodes in this subgraph is equal to that of the given graph. + But it has unconnected subgraphs and moreover total number branches 6= n − 1(= 3). Therefore, it is not a tree.
4
14 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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
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
Network Toplogy
Khublei Shibun!
Thank You! Any Question?
16 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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
2
1
a
b
c
4
(a)
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3
5
3 d
4
(b)
L. Joyprakash Singh (ECE, NEHU)
(c)
EE-304 ENT :: Network topology and graph
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: IL
2 Is
I1
R2
L 3 I2
4 I3
IC
R1
R3
C
R4
Is
1 − +
Vs
5
(a)
18 / 20
L. Joyprakash Singh (ECE, NEHU)
EE-304 ENT :: Network topology and graph
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: IL
2 Is
I1
R2
L 3 I2
2
4
3
4
I3
IC
R1
R3
C
R4
Is
1 − +
Vs
5
(a)
18 / 20
L. Joyprakash Singh (ECE, NEHU)
1, 5
(b)
EE-304 ENT :: Network topology and graph
Home Assignment
Graph and Incidence Matrix
Problems for Practice - II Note: Transformer gives a unconnected graph! For example - 2: I2
1 R1
R2
C 3 IC
2
4 I3
K
I1
I
− +
Vs
R3
6
5
(a)
19 / 20
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! For example - 2: I2
1 R1
R2
C 3 IC
2
4
1
K
R3
I
− + 6
5
(a)
19 / 20
4
I3
I1 Vs
2 3
L. Joyprakash Singh (ECE, NEHU)
6 5
(b)
EE-304 ENT :: Network topology and graph
Home Assignment
Graph and Incidence Matrix
Few non-planar Graphs 1
2
1
3
2
6
3
4 5
(a)
20 / 20
L. Joyprakash Singh (ECE, NEHU)
4
(b)
EE-304 ENT :: Network topology and graph
