Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalPower System

Option A

$n-1$

Quick Summary:

In graph theory and electrical power system network analysis, a tree is defined as a connected subgraph containing all the nodes of the original graph but no closed loops. For a network with $n$ nodes, the number of branches (edges) required to form a tree is exactly $n-1$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$B_t = n - 1$ — where $B_t$ is the number of tree branches and $n$ is the number of nodes

$L = b - n + 1$ — where $L$ is the number of links (co-tree branches) and $b$ is the total branches

⚙️ Working Principle

The principle is based on the Euler's formula for trees. A tree connects all nodes with the minimum number of branches such that adding any additional branch would create a closed loop, and removing any branch would result in an isolated node. Therefore, to connect $n$ vertices without forming a cycle, exactly $n-1$ connections are necessary.

📌 Key Points

▸
A tree must be a connected graph.
▸
A tree contains no closed loops (cycles).
▸
If a graph has $b$ total branches and $n$ nodes, the number of links is $b - (n - 1)$ .
▸
The union of a tree and its links forms the complete graph.

✅ Advantages

▸
Simplifies complex electrical networks for analysis.
▸
Facilitates the formulation of incidence matrices.

❌ Disadvantages / Limitations

▸
Only represents the connectivity, not the physical impedance of the circuit.
▸
Does not account for the nature of the elements (resistor, inductor, capacitor).

🛠️ Applications / Uses

▸
Kirchhoff's Voltage Law (KVL) formulation in network analysis.
▸
Solving complex nodal and mesh equations for large power grids.

📄 Additional Information

▸
This concept is fundamental to the construction of the Tie-set and Cut-set matrices in Electrical Network Topology.
▸
Option B (n+1) is incorrect as it would create at least one closed loop.
▸
Option C (n) is incorrect as it would always result in a graph with at least one cycle.

📊 Diagram / Illustration

✅

A is correct — A tree with $n$ nodes always possesses exactly $n-1$ branches to remain acyclic while ensuring all nodes are connected.

Core Concepts Used

Click any tag to open in AI Tutor

Graph Theory in Electrical Networks Tree and Co-tree Network Topology

💡 EXAM TIP

Always remember that in any connected graph, the total number of branches $b$ equals the sum of tree branches ( $n-1$ ) and links ( $b-n+1$ ).

Discussion (0)

Loading discussion...

Prev Next

Correct Answer

Related Questions

Discussion (0)