Lecture # 1 Introduction to Graph Theory (Network Topology)

Key Concepts:

1. Introduction to Graph Theory/Network Topology

Graph theory, also known as network topology, is a method to represent a circuit consisting of different elements.
A circuit with elements like capacitance, resistance, and inductance can be represented as a graph by considering interconnecting points as nodes.
Example: A circuit with resistors (R1, R2, R3) and an inductor (L1) is transformed into a graph with nodes 1, 2, 3, and 4, where the components are represented as branches connecting these nodes.

2. Basic Terminologies

Branch: Any individual element (e.g., R1, R2, L1) within the graph. The terminals of these branches are called nodes.
Node: The terminals or interconnecting points of branches.
Degree of a Node: The number of branches connected to a particular node.
- Example: If three branches are connected to node 4, the degree of node 4 is 3.
Tree: An interconnected, open (no loops) set of branches that includes all nodes of the graph.
- A given circuit can have multiple possible tree combinations.
- Example: Branches connecting nodes 1-4, 2-4, and 3-4 form a tree because they connect all nodes without creating a closed loop.
Tree Branch (Twig): All the branches that constitute a specific tree.
- Example: If branches 1, 2, and 3 form a tree, then branches 1, 2, and 3 are tree branches (twigs).
Tree Link: The branches in the original graph that are not part of a given tree.
- Example: If branches 1, 2, and 3 are twigs, then the remaining branches in the original graph are links.
Loop: A closed path or contour formed by a combination of branches in the graph.
- Example: Branches connecting nodes 2-3, 3-4, and 4-2 form a loop.

3. Relation Between Twigs and Links

n: Number of nodes in the graph.
Number of Twigs: n - 1
L: Number of links in the graph.
Formula for L: L = B - N - 1 = B - N + 1, where B is the total number of branches in the graph.
Example:
- A graph with 4 nodes (n = 4) will have 4 - 1 = 3 twigs.
- If the graph has 6 total branches (B = 6), then the number of links (L) is 6 - 4 + 1 = 3.

4. Examples of Converting Circuits to Graphs

A single element (e.g., a resistor) in a circuit is represented by two nodes connected by one branch.
Two elements in series are represented by three nodes and two branches.
Two elements in parallel are represented by two nodes and two branches.
A more complex network is broken down into nodes and branches representing the interconnections and components.

5. Oriented Graph (Directed Graph)

An oriented graph, also called a directed graph, is a graph in which a direction (e.g., current flow) is assigned to each branch.
By assigning a direction of current to each branch, the graph becomes an oriented or directed graph.

6. Conclusion

Graph theory provides a method for representing and analyzing electrical circuits using nodes and branches.
Understanding the terminologies (branch, node, tree, link, loop) and their relationships is crucial for applying graph theory to network analysis.
Converting a circuit into a graph allows for the application of graph-theoretic techniques to solve circuit problems.