Graph Theory in Network Analysis | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Network topology is a way to draw electrical circuits so they’re easier to understand. It turns complicated circuits into simple network pictures. It’s also called graph theory, which is a math tool that shows how things connect—like in circuits, social groups, living systems, or computers. Whether it’s power lines, phone networks, or friendships, using graph theory helps make things work better, saves effort, and fixes tough problems.

Graph Theory in Network Analysis

Graph theory in network analysis is about studying graphs, which are like maps showing how things are connected. These graphs use dots (called nodes or vertices) and lines (called edges) to show links between stuff—like computers, parts in a circuit, people, or even living things. It helps us see and understand how everything is tied together.

A graph has:

• Nodes (Vertices): The things in the network, like connection points in a circuit, people in a group, or computers in a system.
• Edges (Links): The lines showing how those things are connected—like wires in a network, friendships, or data links.

Graph theory is great because it makes complicated systems easier to figure out and work with.

Graph: It’s a basic sketch of a circuit or network where all the details are left out, and everything is shown as simple lines.

​Basic Terminology

To understand graph theory, you need to know some basic words it uses. Here are the main ones:

• Node (Vertex): The basic piece of a graph, like a dot that stands for something in the network (like a person or a wire connection).
• Edge (Link): The line that joins two nodes. It can go one way (directed) or both ways (undirected).
• Degree of a Node: How many lines (edges) are attached to a dot (node). It shows how many connections that thing has.
• Path: A chain of edges that link up a bunch of nodes, like a route from one dot to another.
• Cycle: A path that loops back to where it started, making a circle with the same node at the beginning and end.

Types of Graphs 

Graph theory in network analysis uses different kinds of graphs, and each one is good for different things.

• Graph: The basic setup with dots (nodes) and lines (edges). It can have lines that go one way (directed) or both ways (undirected).
• Subgraph: A smaller piece of a graph, with just some of the dots and lines from the bigger one.
• Connected Graph: A graph where you can find a path between any two dots. If it’s not connected, it splits into separate chunks that don’t link up.
• Directed Graph (Oriented Graph): A graph where the lines have a direction, shown as arrows pointing from one dot to another. This is handy for stuff like traffic or sending data.

Tree

A tree is a special kind of graph where all the dots (nodes) are connected, but there are no loops (cycles). Think of it like a family tree: everyone is linked through parents and kids, but you can’t circle back to the same person in a loop. In network analysis, trees are super helpful because they guarantee there’s only one way to travel between any two dots. This is perfect for things like:

• Routing: Imagine planning a delivery route—using a tree means the driver won’t go in circles or take extra roads.
• Simplifying: In a power grid, a tree helps avoid wasteful extra connections.

A tree is a smaller piece of the big graph that still connects all the dots but avoids making any closed loops. If your graph has “n” dots (say, 5 dots), the tree will have n–1 lines (so, 4 lines). Why? Because each new line adds a dot without circling back. For example:

• If you have 3 dots (A, B, C), a tree might be A–B–C (2 lines). Adding a third line, like C–A, would make a loop, so it’s not a tree anymore.

Twig

A twig is just a line (branch) in a tree, and it’s usually drawn thick to stand out. In a tree with n dots, there are always n–1 twigs—same as the number of lines, because every line in a tree is a twig! A twig is a connected piece that often includes “leaf nodes” (dots at the end with only one line attached, like the tips of a real tree’s branches).

In real life:

• Electrical networks: A twig could be a wire connecting a power source to a house. It’s part of the main “tree” of wires delivering electricity.
• Communication: A twig might be a cable linking one phone tower to another in a simple, no-loop setup.

For example, if a tree is A–B–C–D (4 dots, 3 lines), each line (A–B, B–C, C–D) is a twig. The dots at the ends (A and D) are like leaves.

Co-Tree

A co-tree is everything in the graph that’s not part of the tree. It’s made up of the leftover lines (edges) that didn’t get picked to be in the tree. Think of it like this: if the tree is the main road system, the co-tree is all the side paths or shortcuts you didn’t use.

Why care about the co-tree?

• Backup routes: In a computer network, co-tree edges might show extra connections you could use if the main ones fail.
• Redundancy: In a power system, these extra lines might show where you’ve got spare wires that aren’t needed right now.

For example, imagine a graph with 4 dots (A, B, C, D) and edges A–B, B–C, C–D, and A–C. If the tree is A–B–C–D (3 lines), the co-tree is just A–C (the leftover line). That A–C edge could make a loop if added to the tree, so it stays in the co-tree.

Link or Chord

A link (or chord) is a line that connects two dots that aren’t right next to each other in the tree. If you add a link to a tree, it creates a loop (cycle), which is why it’s not part of the tree to begin with. This idea is key for planning networks to keep them efficient and avoid unnecessary connections.

Real-world use:

• Network optimization: If you’re designing a phone network, you might use a tree for the basic setup (no loops), but links show where you could add extra cables—though they’d create overlap.
• Example: In a tree A–B–C, adding a link A–C makes a triangle (loop). That A–C line is the link or chord.

Think of it like a bridge between two far-apart towns. It’s useful but makes the simple “tree” path more complicated.

Degree of a Node

The degree of a node is just a count of how many lines (edges) connect to that dot. It’s a simple way to see how “busy” or “popular” a node is in the network.

• In social networks: If you’re a dot and have 5 friends (5 lines), your degree is 5. A high degree means you’re super connected!
• In circuits: If a junction (dot) has 3 wires attached, its degree is 3. It shows how many parts it’s linked to.

For graphs with arrows (directed graphs), it splits into:

• In-degree: How many arrows point to the dot. Example: If 2 friends message you, your in-degree is 2.
• Out-degree: How many arrows point away from the dot. Example: If you message 3 friends, your out-degree is 3.

For example, in a graph A–B–C where B also connects to D (so it’s A–B–C and B–D), the degrees are:

• A: 1 (just A–B)
• B: 3 (B–A, B–C, B–D)
• C: 1 (C–B)
• D: 1 (D–B)

In a directed version (arrows), if A→B←C→D, then:

• B’s in-degree is 2 (arrows from A and C), out-degree is 1 (to D).

Conclusion

The Network topology and graph theory provide a powerful framework for simplifying and analyzing complex systems across various domains, including electrical circuits, communication networks, and social structures. By representing intricate connections as graphs—comprising nodes, edges, and specialized structures like trees, twigs, co-trees, and links—professionals can efficiently model relationships, optimize designs, and troubleshoot issues. Key concepts such as the degree of a node, paths, cycles, and directed graphs enable precise evaluation of connectivity and flow, ensuring streamlined routing, reduced redundancy, and enhanced system performance. Whether applied to power grids, data transmission, or network planning, graph theory remains an indispensable tool for engineers, analysts, and researchers striving to improve efficiency and solve real-world challenges.