GATE 2018 EE Q.7: solution and answer

GATE 2018 · EEQ.7 · 1 mark · MCQ

The graph of a network has 8 nodes and 5 independent loops. The number of branches of the graph is

(A)
11
(B)
12
(C)
13
(D)
14

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Answer and solution

ANSWER

The graph has 12 branches, which is option (B).

Official answer: BSource: Official answer key, GATE 2018 EE (IIT Guwahati)

Take the network graph as connected, which is the standard assumption for a single network. For a connected graph with $n$ nodes, every tree has $n - 1$ branches (twigs): $n - 1 = 8 - 1 = 7$ .
Every branch not in the tree is a link. Adding a link to the tree closes exactly one fundamental loop, and these loops are independent, so the number of independent loops equals the number of links: $L = b - (n - 1)$
Rearranging for the number of branches: $b = L + n - 1$
Substituting $L = 5$ and $n = 8$ : $b = 5 + 8 - 1 = 12$
Check: $7$ twigs $+$ $5$ links $= 12$ branches. The answer is option (B).

KEY IDEA

In a connected graph, the number of independent loops equals the number of links: $L = b - n + 1$ . A tree uses $n - 1$ branches and each remaining branch closes one fundamental loop, so $b = L + n - 1$ .

WATCH OUT FOR

Using $L = b - n$ , which leaves out the $+ 1$ and gives 13 (option C), or confusing independent loops with meshes or node equations ( $n - 1 = 7$ ). Another mistake is adding $L + n = 13$ without subtracting 1 for the tree.

From the GATE 2018 EE paper. Every question of this paper · Download the full paper · All 25 Laws & node/mesh questions, with notes

GATE 2018 EE Q.7: solution and answer

Answer and solution

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