Unit 1 graph theory

UNIT-1
Network Topology and Graph Theory
1 / 20
prepared by
D V S Ramanjaneyulu
Asst.professor
Ace engineering college

•Network Topology, Terminology,
•Basic cutest and tie set matrices for
planar networks.
•Illustrative Problems,
Syllabus
……

Network Toplogy Terms & Definitions
Circuit
elements:
3 / 20

Circuit elements:
- The mathematical models of a two terminal electrical
devices,
3 / 20

Circuit elements:
-The mathematical models of a two terminal electrical
devices,
-Completely characterized by its voltage-current relationship,
3 / 20

Circuit elements:
devices,
-Can not be subdivided into other two-terminal devices.
3 / 20

Circuit elements:
devices,
Node:
3 / 20

Circuit elements:
-The mathematical models of a two terminal electrical devices,
Node:
-A point at which two or more circuit elements have a
common connection,
3 / 20

Circuit elements:
Node:
common connection,
-The number of branches incident to a node is known as the
degree of that node.
3 / 20

Circuit elements:
Node:
common connection,
Branch:
3 / 20

Circuit elements:
Node:
common connection,
Branch:
-A single path, containing one circuit element, which
connnects one node to any other node,
3 / 20

Circuit elements:
Node:
common connection,
Branch:
-Represented by a line in the graph.
3 / 20

Circuit elements:
Node:
common connection,
Branch:
Path:
3 / 20

Circuit elements:
Node:
-A point at which two or more circuit elements have a common
connection,
Branch:
-A single path, containing one circuit element, which connnects
one node to any other node,
Path:
-A set of elements that may be traversed in order without
passing through the same node twice.
3 / 20

Loop
:
4 / 20

Loop:
- A close path or a closed contour selected in a
network/circuit,
4 / 20

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,
4 / 20

Loop:
node,
-Also known as closed path or circuit.
4 / 20

Loop:
node,
Mesh1 [2]:
4 / 20

Loop:
node,
Mesh1 [2]:
-A loop that does not contain any other loops within it,
4 / 20

Loop:
node,
Mesh1 [2]:
-Any mesh is a circuit/loop but any loop/circuit may not be a
mesh.
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
-The interconnection of two or more circuit elements forms an
electical network.
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
electical network.
Circuit:
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
electical network.
Circuit:
-Network that contains at least one closed path,
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
electical network.
Circuit:
-Every circuit is a network, but not all networks are circuits.
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
electical network.
Circuit:
Planar circuit:
4 / 20

Loop:
node,
Mesh1 [2]:
mesh.
Network:
electical network.
Circuit:
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.
4 / 20

Topology
:
5 / 20

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

Topology:
-Deals with properties of networks which are unaffected when
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

Topology:
and the shape,
nodes are arranged.
Graph:
5 / 20

Topology:
and the shape,
nodes are arranged.
Graph:
-A graph corresponding to a given network is obtained by
replacing all circuit elements with lines.
5 / 20

Topology:
and the shape,
nodes are arranged.
Graph:
- 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

Topology:
and the shape,
nodes are arranged.
Graph:
- 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

Topology:
and the shape,
nodes are arranged.
Graph:
- 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

Network Toplogy Network Circuits & Their Graphs
A circuit with topologically equivalent
graphs:
1
2 IR2
3
IR3
4
−
+
Vs
Is
R1
IR1
R2
R3
C
IC
L
IL
1
2
3
4
1
2
3
4
i) A Circuit ii) its
graph
iii) directed
graph
1
2
3
4
1
2
3
4
1
2
3
4
Three topologically equivalent graphs of figure
ii).
6 / 20

1 2 3
4
5 A
R1 R2
R4
CR3
(a)
1
2
3
4
a b
c
d e f
(b)
Figure
1 :
(a) A circuit and (b) its
graph.
Note:
The maximum number of branches possible, in a circuit, will be
equal to the number of nodes or vertices.
There are at least two branches in a circuit.
7 / 20

1 IR1 2 IR2 3
4
5 A
R1 R2
R4
IR4
C
IC
R3
IR3
(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.
1
2
3
4
a b
c
d e f
8 / 20

1 2 I2
3
I3
4
5
−
+
Vs
g
Is
R
R1
I1
R2
R3
C
IC
L
IL
(a) (b) (c)
1
2
3
4
5
a
b c
fe
d
1
2
3
4, 5
a
b c
fed
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
to simplify analysis and computation.
9 / 20

1
4
−
+Ivs
Vs
R
2 I2
3
R1
I1
R2
Is
C
IC
L
IL
(a) (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.
1
2
3
4
a
b c
ed
1
2
3
4
a
b c
e
d
10 / 20

1 A
1 Ω
1 Ω
1 Ω
1 Ω
+
−
1 Ω
(a) (b) (c)
1 V
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

Tree
:
12 / 20

Tree:
- A connected subgraph having all the nodes of a graph
without any loop.
12 / 20

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

Tree:
any loop.
Twigs:
12 / 20

Tree:
any loop.
Twigs:
-The branches of a tree are known as twigs,
12 / 20

Tree:
any loop.
Twigs:
Links or Chords:
12 / 20

Tree:
any loop.
Twigs:
Links or Chords:
-The branches that are removed from the graph while forming a
tree are termed as links or chords,
12 / 20

Tree:
any loop.
Twigs:
Links or Chords:
-Links are complement of twigs.
12 / 20

Tree:
any loop.
Twigs:
Links or Chords:
Co-tree:
12 / 20

Tree:
any loop.
Twigs:
Links or Chords:
Co-tree:
-The graph constituted with links is known as co-tree.
12 / 20

Given a Graph:
1 a 2 b 3
4
c
d e
f
13 / 20
Tree Twigs of tree Links of cotree
1
a
c
2 b 3
d e
f
4
{a,b,d} {c,e,f}
1
a
c
2 b 3
d e
f
4
{a,d,f} {c,b,e}

Q. Does the following graph with branches a and e form a tree?
1
2
3
4
a
b c
fed
14 / 20

1
2
3
4
a
b c
fed
☞ The number of nodes in this subgraph is
equal to that of the given graph.
14 / 20

1
2
3
4
a
b c
e fd
☞ 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.
14 / 20

Network Toplogy 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

Network Toplogy Khublei Shibun!
Thank You!
Any Question?
16 / 20

Home Assignment 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 3
4
(a)
a
b c
d
(b)
3
4
5
(c)
17 / 20

Note: While replacing all elements of the network with lines to form
a graph, we replace active elements by their internal
resistances to simplify analysis and computation.
For example - 1:
1
Vs
5
(a)
2
Is
3 I2
4
−
+
R1
R2
I1
R3
R4
I3
C
IC
L
IL
Is
18 / 20

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:
5
−
+
1
Vs
2
Is
3 I2
4
R1
R2
I1
R3
R4
I3
C
IC
L
IL
Is
(a)
2
3
4
1, 5
(b)
18 / 20

Note: Transformer gives a unconnected
graph!
For example - 2:
1
2 4
56
−
+
Vs
R1
I1
K
3 IC
R2
I2
C
R3
I3
I
(a)
19 / 20

Note: Transformer gives a unconnected
graph!
For example - 2:
1
2
56
−
+
3 IC
4
Vs
R1
I1
R2
I2
K
C
R3
I3
I
(a)
1
2 3
6 5
4
(b)
19 / 20

1
2
3
4
(a)
1 2
3
45
6
(b)
20 / 20

Unit 1 graph theory

More Related Content

What's hot (20)

Similar to Unit 1 graph theory (20)

More from ACE ENGINEERING COLLEGE (18)

Recently uploaded (20)

Unit 1 graph theory

Editor's Notes