UNIT-2
Network Topology
Q.1) Explain briefly trees, co-trees, and loops in a graph of network with suitable example.
JAN.2015, JUNE 2014
Ans. : Tree : Tree is a set of branches with every node connected to every other node, such that
any one of the branches removed changes this property. In other words, we can state that it is a
connected sub graph of a connected graph containing all the nodes of the graph not forming any
loop. Thus, tree is a set of branches with all nodes not forming any loop or closed path. A
graph and some of the possible trees are as shown in Fig.
Properties of tree :
(1) Tree contains all nodes on the graph.
(2) Tree does not contain any closed path.
(3) In a tree, there exists only one path between any pair of nodes.
(4) In a tree, maximum end nodes of terminal nodes are two.
(5) Every connected graph has at least one tree.
(6) The rank of the tree is same as the rank of graph i.e. (n-1).
(7) Tree contain (n-1) branches if n are the nodes of the tree.
Cotrees :A set of branches forming a complement of tree is called cotree. The number of
branches of a cotree equals b-(n-1) where b is number of branches of a graph.
Loops : In general, a network consists of ‘n’ nodes which are interconnected in some way by ‘b’
branches, then it is possible to transverse adjacent branches starting at any node and return to the
original starting node in different ways. Such a closed path formed by the network branches is
called as loop.
Loop is a connected subgraph of a connected graph such that at each node there are two branches
incident. If the two terminals are made to coincide, it will generate a loop or circuit. The following
Fig shows different loops.
Properties of Loops : A loop of a graph has following properties.
(1) There are exactly two paths between any pair of nodes in the circuit.
(2) There exist at least two branches in loop.
(3) The maximum possible branches in a loop are equal of nodes.
Q.2) For the network shown in Fig. draw its dual. Write in inter go differential form
i) mesh equations for the given network
ii) node equations for the dual. V(t) = 10 sin 40t
JUNE 2015
Sol. : In the given network placing a dot in three independent meshes and assuming datum node
outside the network as shown in the fig. 4(a). the dual network can be drawn by tracing every
element in the meshes. Hence the dual network is as shown in the fig. 4(b).
Q.3) Construct a tree for the network shown in fig. so that all loop currents pass through 7 Ω .
Write the corresponding tie set matrix.
JUNE 2014
Sol.: For the given network, the oriented graph can be drawn by assuming random Orientations in
various branches. Let the random orientations be as shown in the fig. (A). While drawing oriented
graph, the current sources are open circuited as shown in the fig. (B). Let a to e be the nodes while
1 through 7 be the branches of the oriented graph .
Some of the trees of a graph with all the loop currents pass through 7Ω are as shown in the fig.
(c) through (f).
Some of the trees of a graph with all the loop currents pass through 7Ω are as shown in the fig.
7(c) through (f).
Some of the trees of a graph with all the loop currents pass through 7Ω are as shown in the fig.
7(c) through (f).
Q.4) What are dual networks ? what is their significance ? Draw the dual of the circuit shown in
fig
Sol.: Dual Networks.:
JUNE 2015, JAN.2014
Two networks are said to be dual networks of each other if the mesh (loop) Equations of given
network are the node equations of other network.
Significance of dual network:
The dual network can be drawn for ac as well as dc circuits as well as for
resistances and impedances . To get a dual network , put a dot in each independent loop and place
a datum node outside network as shown in the fig. (a). Then trace the elements in each loop from
to the dots placed in each loop . Then trace common elements between
the tree loops. The dual network of the given network is as shown in the fig . (b).
Q. 5) Define the following terms with reference to network topology. Give examples.
i) Tree ii) Graph iii) Sub-graph iv) Tie-set v) Cut-set
JUNE 2014
Graph:
In the given network if all the branches are represented by line segments then the resulting figure
is called the graph of a network (or linear graph). The internal impedance of an ideal voltage
source is zero and hence it is replaced by a short circuit and that of an ideal current source is
infinity and hence it is represented by an open circuit in the graph. Node
Branch (or Twig):
It is a path directly joining two nodes. There may be several parallel paths between two nodes.
Oriented Graph
If directions of currents are marked in all the branches of a graph then it is called an oriented (or
directed) graph.
Connected graph
A network graph is connected if there is a path between any two nodes. Let us assume that the
graph is connected. Since, if it is not connected each disjoint part may be analysed separately as a
connected graph.
Unconnected graph
If there is no path between any two nodes,then the graph is called an unconnected graph.
Planar graph
A planar graph is a graph drawn on a two dimensional plane so that no two branches intersect at
a point which is not a node.
Non – planar graph
A graph on a two – dimensional plane such that two or more branches intersect at a point other
than node on a graph.
Tree of a graph
Tree is a set of branches with all nodes not forming any loop or closed path.
(*) Contains all the nodes of the given network or all the nodes of the graph
(*) No closed path
(*) Number of branches in a tree = n-1 , where n=number of nodes
Co- tree
A Co- tree is a set of branches which are removed so as to form a tree or in other words, a cotree
is a set of branches which when added to the tree gives the complete graph. Each branch so
removed is called a link. Number of links = l = b – (n-1) where b = Total number of branches
n = Number of nodes