Network Topology Questions and Answers

Q: In the graph shown in the following figure, the number of trees (P) and the number of cut sets (Q) are :

The graph has P = 4 trees and Q = 6 cut sets.The two diagonals cross without a node, so the graph has 4 nodes and 4 branches. Tracing it: top left to top right (1), top right to bottom left (3), bottom left to bottom right (4), and bottom right back to top left (2). It is a single closed loop of four branches.A tree joins all 4 nodes with 4 - 1 = 3 branches and no loop. Removing any one of the four branches leaves such a tree, so there are 4 trees.A cut set is a minimal set of branches whose removal splits the graph into two parts. In a single loop, removing one branch leaves it connected, but removing any two branches separates it, so every pair is a cut set: \(\binom{4}{2} = 6\).

Q: If a network contains B branches and N nodes, then the number of mesh current equations would be :

The number of mesh equations is B − (N − 1).For a connected planar network, the number of independent loops (meshes) equals the number of links, the branches not in a tree. A tree joining N nodes has N − 1 branches, so:\[ \begin{aligned} \text{meshes} &= B - (N - 1) \\ &= B - N + 1 \end{aligned} \]For example, a network with 5 branches and 4 nodes needs 2 mesh equations. N − 1 itself is the number of independent node equations. B − N − 1 has the wrong sign on the 1, and the other two forms have no basis in network topology.

Q: A network has 10 nodes and 17 branches. The number of different node pair voltages would be :

The number of node pair voltages is 45.A node pair voltage is the voltage between any two nodes, so the count is the number of ways of choosing 2 nodes out of 10:\[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]The number of branches does not enter this count. It matters for other quantities: the number of independent loops is b - n + 1 = 17 - 10 + 1 = 8, and the number of independent node voltages is n - 1 = 9. Seven is b - n, 17 is the branch count, and 50 does not follow from either figure.

Find the number of branches and nodes in the given circuit

Circuit: an 18 V source, a resistor RA and a 6 ohm resistor (carrying current ix) in parallel on the left; a 5 ohm resistor carrying 13 A in the top wire to a voltage source Vx on the right

A.03 branches, 05 nodes
B.05 branches, 03 nodes
C.05 branches, 05 nodes
D.03 branches, 03 nodes

Exam Year(s)

DSSSB AE - 2023,

Answer: B

In circuit theory a branch is a single element (or a series group of elements carrying the same current) between two nodes, and a node is a point where two or more elements join. Reading the figure: the bottom wire is one node, the top left wire where the 18 V source, \(R_A\) and the 6 ohm resistor meet is a second node, and the junction between the 5 ohm resistor and the source \(V_x\) is a third. So there are three nodes.

The elements are the 18 V source, \(R_A\), the 6 ohm resistor, the 5 ohm resistor and \(V_x\), five elements, each in its own branch, so there are five branches. The count is 5 branches and 3 nodes, option B.

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In the graph shown in the following figure, the number of trees (P) and the number of cut sets (Q) are :

Four nodes at the corners of a rectangle; branch 1 joins the top two, branch 4 the bottom two, and branches 2 and 3 are crossing diagonals with no node where they cross

A.P = 2, Q = 2
B.P = 2, Q = 6
C.P = 4, Q = 6
D.P = 4, Q = 10

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

Answer: C

The graph has P = 4 trees and Q = 6 cut sets.

The two diagonals cross without a node, so the graph has 4 nodes and 4 branches. Tracing it: top left to top right (1), top right to bottom left (3), bottom left to bottom right (4), and bottom right back to top left (2). It is a single closed loop of four branches.

A tree joins all 4 nodes with 4 - 1 = 3 branches and no loop. Removing any one of the four branches leaves such a tree, so there are 4 trees.

A cut set is a minimal set of branches whose removal splits the graph into two parts. In a single loop, removing one branch leaves it connected, but removing any two branches separates it, so every pair is a cut set: \(\binom{4}{2} = 6\).

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If a network contains B branches and N nodes, then the number of mesh current equations would be :

A.B − (N − 1)
B.N − (B − 1)
C.B − N − 1
D.(B + N) − 1

Exam Year(s)

MH SET - 2019, SET Exam - 2019,

Answer: A

The number of mesh equations is B − (N − 1).

For a connected planar network, the number of independent loops (meshes) equals the number of links, the branches not in a tree. A tree joining N nodes has N − 1 branches, so:

\[ \begin{aligned} \text{meshes} &= B - (N - 1) \\ &= B - N + 1 \end{aligned} \]

For example, a network with 5 branches and 4 nodes needs 2 mesh equations. N − 1 itself is the number of independent node equations. B − N − 1 has the wrong sign on the 1, and the other two forms have no basis in network topology.

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A network has 10 nodes and 17 branches. The number of different node pair voltages would be :

A.7
B.17
C.45
D.50

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The number of node pair voltages is 45.

A node pair voltage is the voltage between any two nodes, so the count is the number of ways of choosing 2 nodes out of 10:

\[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]

The number of branches does not enter this count. It matters for other quantities: the number of independent loops is b - n + 1 = 17 - 10 + 1 = 8, and the number of independent node voltages is n - 1 = 9. Seven is b - n, 17 is the branch count, and 50 does not follow from either figure.

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