Graph Theory in Electrical Networks: Branch, Node, Tree, Link, Loop and Cut-Set

Search Description: Learn graph theory in electrical networks including branch, node, graph, path, tree, twig, co-tree, link, loop, tie-set and cut-set in simple language.

Introduction

Graph theory is an important part of electrical network analysis. It helps us understand how different circuit elements are connected without focusing first on their values such as resistance, inductance or capacitance.

In network theory, a circuit can be represented as a graph made of branches and nodes. This graphical method is very useful for forming loop equations, node equations, tie-set matrices and cut-set matrices.

What is a Branch?

A branch represents one circuit element. It is a line joining two nodes in a network graph. A resistor, capacitor, inductor, voltage source or current source can be represented as a branch.

In simple words, every element of a circuit is usually shown as one branch in the graph.

What is a Node?

A node is a common point where two or more branches meet together. In electrical circuits, a node may represent a junction point where current enters or leaves.

Nodes are very important in nodal analysis because node voltages are used to write circuit equations.

Difference Between Branch and Node

Branch	Node
Represents a circuit element	Represents a connection point
Connects two nodes	Connects two or more branches
Example: resistor branch	Example: junction point

What is a Graph in Network Theory?

A graph is a collection of nodes and branches. It shows the geometrical interconnection of circuit elements but does not show the actual physical size or layout of the circuit.

A graph is called a connected graph if there is a path directly or indirectly between every pair of nodes.

Directed and Undirected Graph

Directed Graph

If every branch of a graph has an arrow showing its reference direction or orientation, it is called a directed graph or oriented graph.

Undirected Graph

If the branches do not have any assigned direction, the graph is called an undirected graph.

Properties of a Circuit in a Graph

• There are at least two branches in a circuit.
• There are exactly two paths between any pair of nodes in a circuit.
• The maximum number of branches in a circuit may be equal to the number of nodes in the graph.

What is a Path?

A path is a sequence of connected branches through which we can move from one node to another without repeating branches unnecessarily.

A path has the following properties:

1. At the terminal nodes, only one branch is incident.
2. At the remaining intermediate nodes, two branches are incident.

What is a Tree?

A tree is a connected subgraph that contains all the nodes of the original graph but does not contain any closed path or loop.

In simple words, a tree connects all the nodes of the graph using the minimum number of branches without forming any loop.

Number of twigs = n − 1

Where n is the number of nodes in the graph.

What are Twigs?

The branches present in a tree are called twigs. Since a tree contains all nodes and no closed path, the number of twigs is always equal to n − 1.

What is a Co-Tree?

The branches of the original graph that are not included in the tree form the co-tree. The co-tree is also called the complement of the tree.

What are Links or Chords?

The branches that belong to the co-tree are called links or chords.

Number of links = b − n + 1

Where:

• b = number of branches
• n = number of nodes

Tree and Co-Tree Relationship

A graph is the union of its tree and co-tree. This means:

Graph = Tree + Co-tree

For a given graph, many different trees can be drawn depending on the selected branches. Therefore, decomposition of a graph into tree and co-tree is not unique.

Properties of a Tree

1. A tree contains all the nodes of the graph.
2. A tree does not contain any closed path.
3. Different trees are possible for the same graph.
4. If a graph has n nodes, the tree has n − 1 branches.
5. The remaining branches form the co-tree.
6. The number of link branches is b − n + 1.

What is a Loop or Circuit?

A loop is a closed path in a graph. If a link is added to a tree, the resulting graph contains exactly one closed path. This closed path is called a loop or circuit.

Properties of a Loop

1. There are exactly two paths between any pair of nodes in a loop.
2. There must be at least two branches in a loop.
3. The maximum possible number of branches in a loop may be equal to the number of nodes in the graph.

What is a Fundamental Loop or Tie-Set?

A loop that contains only one link is called a fundamental loop, f-loop or tie-set.

The number of fundamental loops is equal to the number of links.

Number of fundamental loops = b − n + 1

The orientation of a fundamental loop is usually chosen in the same direction as its link.

What is a Cut-Set?

A cut-set is a minimum set of branches that, when removed from a connected graph, separates the graph into two connected subgraphs.

In simple words, removing a cut-set disconnects the original network into two parts.

What is a Fundamental Cut-Set?

A cut-set that contains only one twig is called a fundamental cut-set or f-cut-set.

The number of fundamental cut-sets is equal to the number of twigs.

Number of fundamental cut-sets = n − 1

The orientation of a fundamental cut-set is chosen in the same direction as its twig.

Important Formulas in Network Graph Theory

Quantity	Formula
Number of twigs	nt = n − 1
Number of links	nl = b − n + 1
Number of fundamental loops	b − n + 1
Number of fundamental cut-sets	n − 1

Difference Between Tree, Twig, Link and Co-Tree

Term	Meaning
Tree	Connected subgraph containing all nodes and no loop
Twig	Branch that belongs to a tree
Co-tree	Set of branches not included in the tree
Link / Chord	Branch that belongs to the co-tree

Applications of Graph Theory in Electrical Engineering

• Network analysis
• Formation of KCL and KVL equations
• Tie-set matrix formation
• Cut-set matrix formation
• Mesh analysis
• Nodal analysis
• Power system network modeling
• Computer-aided circuit analysis
• Network topology study

Beginner Notes

• A branch represents a circuit element.
• A node is a junction where branches meet.
• A graph shows circuit connectivity.
• A tree connects all nodes without forming a loop.
• Tree branches are called twigs.
• Branches outside the tree are called links.
• Adding one link to a tree forms one fundamental loop.
• Removing one twig forms one fundamental cut-set.

Frequently Asked Questions

What is a branch in network graph theory?

A branch is a line that represents a circuit element and connects two nodes.

What is a node?

A node is a common point where two or more branches meet.

What is a tree in network theory?

A tree is a connected subgraph that includes all nodes of the graph but does not contain any closed path.

What is the number of twigs in a graph?

If a graph has n nodes, then the number of twigs is n − 1.

What is the number of links in a graph?

If a graph has b branches and n nodes, then the number of links is b − n + 1.

What is a cut-set?

A cut-set is a minimum set of branches which, when removed, separates a connected graph into two connected subgraphs.

Conclusion

Graph theory provides a clear way to understand the structure of electrical networks. Terms like branch, node, graph, path, tree, twig, link, loop and cut-set are essential for advanced network analysis.

For beginners, the most important formulas to remember are: number of twigs = n − 1 and number of links = b − n + 1. These formulas are used in fundamental loop and cut-set analysis.