Question

The number of nodes in a graph having 5 branches and 2 independent loops is

• A. \(3 \)
• B. \(4 \)
• C. \(5 \)
• D. \(6 \)

Hint:

Always remember the fundamental graph theory relationship: Loops = Branches - Nodes + 1. Rearranging it gives Nodes = Branches - Loops + 1, which simplifies finding parameters instantly.

Answer 1

The Correct Option is B

Understanding the Concept: In network topology and graph theory applied to electrical circuits, the fundamental relationship between the number of branches (\(b\)), the number of nodes (\(n\)), and the number of independent loops or fundamental loops (\(l\)) is governed by Euler’s formula for graphs. This structural equation is explicitly written as: \[ l = b - n + 1 \] Where:
• \(b\) represents the total number of branches in the network graph.
• \(n\) represents the total number of junctions or nodes where two or more elements connect.
• \(l\) represents the number of independent meshes or loops contained within the graph. By rearranging this foundational equation, we can directly determine any one of the variables if the other two parameters are known.

Step 1: Extracting the given parameters from the problem statement.
From the question, we are given the following explicit topological properties of the graph:
• Total number of branches, \(b = 5\)
• Total number of independent loops, \(l = 2\)

Step 2: Substituting the values into the formula to find the number of nodes (\(n\)).
We begin with the standard network topology relationship: \[ l = b - n + 1 \] To isolate the variable representing the number of nodes (\(n\)), we add \(n\) to both sides of the equation and subtract \(l\) from both sides: \[ n = b - l + 1 \] Now, substitute the given numerical values of \(b = 5\) and \(l = 2\) directly into this rearranged expression: \[ n = 5 - 2 + 1 \] Performing the simple arithmetic operations step by step: \[ 5 - 2 = 3 \] \[ 3 + 1 = 4 \] Thus, the total number of nodes present in the given graph is exactly equal to 4. This matches perfectly with Option (B).