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Transcript of Electrical Network, Graph Theory, Incidence Matrix, Topology
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7/29/2019 Electrical Network, Graph Theory, Incidence Matrix, Topology
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Incidence Matrix, AiEE-304 ENT credits: 4 L{3} P{0} T{1}
Lairenlakpam Joyprakash Singh, PhD
Department of ECE,
North-Eastern Hill University (NEHU),Shillong 793 [email protected]
August 8, 2013
1 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
1 Which branches are incident at which nodes,
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
1 Which branches are incident at which nodes,2 What are the orientations of branches relative to the nodes.
2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
G
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
1 Which branches are incident at which nodes,2 What are the orientations of branches relative to the nodes.
Types:
2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
G T I M
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
1 Which branches are incident at which nodes,2 What are the orientations of branches relative to the nodes.
Types:
1 Complete incidence matrix, Ai,
2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Graph Theory Incidence Matrix
Graph Theory: Incidence Matrix
Incidence matrix provides information like
1 Which branches are incident at which nodes,2 What are the orientations of branches relative to the nodes.
Types:
1 Complete incidence matrix, Ai,
2 Reduced incidence matrix, A.
2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Graph Theory Incidence Matrix
Complete Incidence Matrix, Ai
A complete incidence matrix of a connected graph with 4 branchesand 5 nodes is given by:
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a24
3 a31 a32 a33 a344 a41 a42 a43 a445 a51 a52 a53 a54
3 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Graph Theory Incidence Matrix
Complete Incidence Matrix, Ai
A complete incidence matrix of a connected graph with 4 branchesand 5 nodes is given by:
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a24
3 a31 a32 a33 a344 a41 a42 a43 a445 a51 a52 a53 a54
In a matrix, Ai, with n rows and b columns, an entry, aij , in the ith
row and jth column has the following values:
aij =
1, if the branch j is incident to and oriented away from the node, i,1, if the branch j is incident to and oriented towards the node, i,
0, if the branch j is not incident to the node, i.
3 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Graph Theory Incidence Matrix
Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
a
b
cd
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
a
b
cd
The complete incidence matrix of the above graph with matrixelement values is
branches
nodes
a b c d
Ai =
1
2
3
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
aa
b
cd
The complete incidence matrix of the above graph with matrixelement values is
branchesnodes
a b c d
Ai =
1
2
3
1
0
1
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
a
bb
cd
The complete incidence matrix of the above graph with matrixelement values is
branchesnodes
a b c d
Ai =
1
2
3
1 1
0 1
1 0
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
a
b
ccd
The complete incidence matrix of the above graph with matrixelement values is
branchesnodes
a b c d
Ai =
1
2
3
1 1 0
0 1 1
1 0 1
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Graph Theory Incidence Matrix
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Complete Incidence Matrix, Ai
An complete incidence matrix of a graph with n = 3 and b = 4
Ai =
branchesnodes
a b c d
1 a11 a12 a13 a142 a21 a22 a23 a243 a31 a32 a23 a24
1 2
3
a
b
cdd
The complete incidence matrix of the above graph with matrixelement values is
branchesnodes
a b c d
Ai =
1
2
3
1 1 0 1
0 1 1 0
1 0 1 1
4 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b c
d e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b c
d e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
aa
b c
d e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1
0
1
0
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
bb c
d e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1 1
0 1
1 0
0 0
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b cc
d e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1 1 0
0 1 1
1 0 1
0 0 0
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
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Example - I
1 2 3
4
+Vs
Is
R
R1I1 R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b c
dd e f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1 1 0 1
0 1 1 0
1 0 1 0
0 0 0 1
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
E l I
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Example - I
1 2 3
4
+Vs
Is
R
R1
I1 R2
I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b c
d ee f
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1 1 0 1 0
0 1 1 0 1
1 0 1 0 0
0 0 0 1 1
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
E l I
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Example - I
1 2 3
4
+Vs
Is
R
R1I1
R2I2
R3
I3
C
IC
LIL
(a)
12
3
4
a
b c
d e ff
(b)
Figure 1 : A circuit (a) and its graph (b).
The complete incidence matrix of the above graph is
Ai =
1 1 0 1 0 0
0 1 1 0 1 0
1 0 1 0 0 1
0 0 0 1 1 1
5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
I id M t i A d A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
I id M t i A d A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
I id M t i A d A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Incidence Matrices A and A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
iii) The determinant ofAi of a closed loop is zero.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Incidence Matrices: A and A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
iii) The determinant ofAi of a closed loop is zero.
Incidence matrix or Reduced incidence matrix, A:
1 If any row is removed from the incidence matrix Ai with nxb dimension, theremaining matrix is known as a reduced incidence matrix, A.Where n is the total number of nodes and b the total number of branches inthe given graph.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Incidence Matrices: A and A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
iii) The determinant ofAi of a closed loop is zero.
Incidence matrix or Reduced incidence matrix, A:
1 If any row is removed from the incidence matrix Ai with nxb dimension, theremaining matrix is known as a reduced incidence matrix, A.Where n is the total number of nodes and b the total number of branches inthe given graph.
2 The order of the reduced incidence matrix A is (n 1) b, i.e. it has (n 1)rows and b columns.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Incidence Matrices: Ai and A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
iii) The determinant ofAi of a closed loop is zero.
Incidence matrix or Reduced incidence matrix, A:
1 If any row is removed from the incidence matrix Ai with nxb dimension, theremaining matrix is known as a reduced incidence matrix, A.Where n is the total number of nodes and b the total number of branches inthe given graph.
2 The order of the reduced incidence matrix A is (n 1) b, i.e. it has (n 1)rows and b columns.
3 Normally, the row corresponding to the reference node is deleted to form areduced incidence matrix, A, from an incidence matrix, Ai.
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Incidence Matrices: Ai and A
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Incidence Matrices: Ai and A
Complete incidence matrix, Ai, has the order n b and satifies the followingproperties:
i) The algebraic sum of elements in any column ofAi is zero.ii) Given the incidence matrix A, the corresponding graph can easily be
constructed since Ai is a complete mathematical replica of the graph.
iii) The determinant ofAi of a closed loop is zero.
Incidence matrix or Reduced incidence matrix, A:
1 If any row is removed from the incidence matrix Ai with nxb dimension, theremaining matrix is known as a reduced incidence matrix, A.Where n is the total number of nodes and b the total number of branches inthe given graph.
2 The order of the reduced incidence matrix A is (n 1) b, i.e. it has (n 1)rows and b columns.
3 Normally, the row corresponding to the reference node is deleted to form areduced incidence matrix, A, from an incidence matrix, Ai.
Ai =
1 1 0 1 0 00 1 1 0 1 0
1 0 1 0 0 10 0 0 1 1 1
A =
1 1 0 1 0 00 1 1 0 1 0
1 0 1 0 0 1
Incidence Matrix, Ai Reduced incidence matrix, A
6 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Network, Graph, Ai, and A
Number of possible trees in the graph Fig 1(b)
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Number of possible trees in the graph, Fig. 1(b)Let us considered the Figure 1 and write the incidence matrix, Ai and its reduced incidence matrix,A as
Ai =
1 1 0 1 0 00 1 1 0 1 0
1 0 1 0 0 10 0 0 1 1 1
A = 1 1 0 1 0 0
0 1 1 0 1 01 0 1 0 0 1
The transpose of the reduced incidence matrix, A is then given by
AT =
1 0 11 1 00 1 1
1 0 00 1 0
0 0 1
And now we have,
AAT =
1 1 0 1 0 00 1 1 0 1 0
1 0 1 0 0 1
1 0 11 1 00 1 1
1 0 00 1 00 0 1
=
3 1 11 3 11 1 3
Therefore the possible number of trees that can be constructed from the graph in Figure 1(b) is
AAT = 3
3 1
1 3
(1)1 11 3
+ (1)1 31 1
= 3(9 1) + (3 1) 1(1 + 3)
Although 6C3 = 20 combinations exist for tree formation, the p ossible combnations areAAT = 24 4 4 = 16 only.
7 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Possible trees of a network
Drawing all possible trees of the network shown in 1(a)
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Drawing all possible trees of the network shown in 1(a)
1 2 3
4
+
Vs
Is
R
R1
I1
R2
I2
R3
I3
C
IC
LIL
(a) A Circuit
12
3
4
a
b c
fed
(b) The graph
Possible trees of the above graph [Trees 1 - 4]:
12
3
4
a
b c
fed
(c) Tree 1
12
3
4
a
b c
fed
(d) Tree 2
12
3
4
a
b c
fed
(e) Tree 3
12
3
4
a
b c
fed
(f) Tree 4
8 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Possible trees of a network
Drawing all possible trees
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Drawing all possible trees . . .
Possible trees of the given graph [Trees 5 - 10]:
12
3
4
a
b c
fed
(g) Tree 5
12
3
4
a
b c
fed
(h) Tree 6
12
3
4
a
b c
fed
(i) Tree 7
12
3
4
a
b c
fed
(j) Tree 8
12
3
4
a
b c
fed
(k) Tree 9
12
3
4
a
b c
fed
(l) Tree 10
9 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Possible trees of a network
Drawing all possible trees . . .
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Drawing all possible trees . . .
Possible trees of the given graph [Trees 11 - 16]:
12
3
4
a
b c
fed
(m) Tree 11
12
3
4
a
b c
fed
(n) Tree 12
12
3
4
a
b c
fed
(o) Tree 13
12
3
4
a
b c
fed
(p) Tree 14
12
3
4
a
b c
fed
(q) Tree 15
12
3
4
a
b c
fed
(r) Tree 16
1 0 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Possible trees of a network
Summary:
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Summary:
In the given graph, we have the number of nodes and the branches are n = 4 and b = 6 respectively.Hence, total number of twigs and links in a tree are, t = n 1 = 3, and l = b t = b n + 1 = 3.Possible trees of the given graph Fig. 1(b) are only:
Possible trees(16 Nos.) drawn above may be written as:
Tree 1 : Twigs{a, b, d},Links{c, e, f}Tree 2 : Twigs{a, b, e},Links{c, d, f}Tree 3 : Twigs{a, c, e},Links{b, d, f}Tree 4 : Twigs{a, c, f},Links{b, d, e}Tree 5 : Twigs{a, b, f},Links{c, d, e}
.
.
.
.
.
.
.
.
.Tree 16 :Twigs{b, e, f},Links{a, c, d}
1 1 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 00 0 0 1 1 1
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
OR
1 2
3 4
a
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b
OR
1 2
3 4
a
b
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Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b c
OR
1 2
3 4
a
bc
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b c
dOR
1 2
3 4
a
b
c
d
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b c
d eOR
1 2
3 4
a
b
c
d
e
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 01 0 1 0 0 1
0 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 0
1 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b c
d e fOR
1 2
3 4
a
b
c
d
e
f
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph
Exercise:
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Q. Draw a complete graph from the following incidence matrix.
A =
1 1 0 1 0 0
1 0 1 0 0 10 1 1 0 1 0
Answer: Creating a complete Incidence Matrix, Ai, from the givenreduced incidence matrix, A, we get
Ai =
1 1 0 1 0 0
1 0 1 0 0 10 1 1 0 1 00 0 0 1 1 1
And its graph
1 23
4
a
b c
d e fOR
1 2
3 4
a
b
c
d
e
f
1 2 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Incidence Matrix and KCL
Incidence Matrix and KCL
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Incidence Matrix and KCL:
Kirchhoffs current law (KCL) of a graph can be expressed in terms of
the reduced incidence matrix as AiIb = 0 where Ib represents branchcurrent vectors I1, I2, I3, I4, I5, and I6.
1 3 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Incidence Matrix and KCL
Incidence Matrix and KCL
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Incidence Matrix and KCL:
Kirchhoffs current law (KCL) of a graph can be expressed in terms of
the reduced incidence matrix as AiIb = 0 where Ib represents branchcurrent vectors I1, I2, I3, I4, I5, and I6.
For example:
1 0 1 0 1 0 0 01 1 0 0 0 0 0 1
0 0 1 1 0 1 0 00 1 0 1 0 0 1 0
I1I2
I3I4I5I6
=
0
000
1 3 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph Incidence Matrix and KCL
Incidence Matrix and KCL
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A complete incidence matrix Ai and a graph constructedfrom it are given below
Ai =
1 0 1 0 1 0 0 0
1 1 0 0 0 0 0 10 0 1 1 0 1 0 00 1 0 1 0 0 1 00 0 0 0 1 1 1 1
If branch currents of branches a,b,c,d,e,f,g,h areI1, I2, I3, I4, I5, I6, I7 and I8 respectively, then usingAIb = 0 we have
1
2
34
5
ab
cd
e f g
h
1 0 1 0 1 0 0 01 1 0 0 0 0 0 1
0 0 1 1 0 1 0 00 1 0 1 0 0 1 0
I1
I2I3I4I5I6I7I8
=
0000
Branch current equations: from above equation- after applying KCL at nodes 1, 2, 3 and 4in the graph-
I1 + I3 I5 = 0I1 + I2 + I8 = 0I3 + I4 + I6 = 0I2 I4 + I7 = 0
I1 + I3 I5 = 0I1 + I2 + I8 = 0I3 + I4 + I6 = 0I2 I4 + I7 = 0
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Network & Graph Incidence Matrix and KCL
Incidence Matrix and Branch Voltages
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Incidence Matrix and Branch Voltages:If branch voltages of branches a, b, c, d, e, f, g and h in the ciruit areVa, Vb, Vc, Vd, Ve, Vf, Vg and Vh while nodes-to-reference voltages aree1, e2, e3, e4, e5, e6, e7 and e8 respectively, then the branch voltagevector, V, and the node voltage vector e are given by
V =
VaVbVcVdVe
VfVgVh
and e =
e1e2e3e4e5
e6e7e8
Then branch voltages of the last graph are then given by [AT][e] = [V]
1 1 0 00 1 0 11 0 1 00 0 1 1
1 0 0 00 0 1 00 0 0 10 1 0 0
e1e2e3e
4e5e6e7e8
=
VaVbVcVdVeVfVgVh
1 5 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
Network & Graph 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.
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Network & Graph 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. EdministerSchuams 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 & Graph Khublei Shibun!
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Thank You!
Any Question?
1 7 / 1 7 L. Joyprakash Singh (ECE, NEHU) EE-304 ENT :: Incidence Matrix
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