The best examples of introduction to graph theory with practical examples

Before any definitions, let’s walk through examples of introduction to graph theory with practical examples that you’ve probably seen without realizing it.

Think about these situations:

You’re using a subway map to plan the fastest way across a city. Each station is a point, and each track between stations is a connection. That’s a graph.

You’re on a social media platform looking at mutual friends. Each person is a point, and each friendship is a connection between two points. That’s a graph.

A delivery company wants to plan routes so a driver can visit many addresses without wasting time and gas. Each address is a point, and the roads between them are connections. Again, a graph.

These are all examples of introduction to graph theory with practical examples in disguise: real-world systems turned into points (called vertices) and connections (called edges) so we can reason about them more clearly.

A gentle definition: graphs in plain language

Now that you’ve seen a few real examples, let’s put some names to what you already understand.

A graph is just:

• A set of vertices (also called nodes): the “things” in your system
• A set of edges: the “relationships” or connections between those things

If you think of a city map:

• Intersections or locations are vertices
• Roads between them are edges

If you think of a social network:

• People are vertices
• Friendships or follows are edges

In some graphs, edges have directions (like one-way streets or “A follows B” on social media). These are directed graphs. In others, connections go both ways (like a mutual friendship). Those are undirected graphs.

Edges can also have weights: numbers that measure something about the connection—distance, time, cost, risk, or capacity. A road might have a weight equal to its travel time in minutes. A network cable might have a weight equal to its bandwidth.

So when you see the phrase examples of introduction to graph theory with practical examples, you can translate it as: “Real situations where we can model things as vertices and edges and actually do something useful with that model.”

Everyday examples include maps, routes, and navigation apps

Let’s start with one of the best examples because almost everyone uses it daily: navigation.

When you open a navigation app and ask for the fastest route, you’re asking it to solve a shortest path problem on a graph.

• Vertices: intersections, highway exits, or key points along the road network
• Edges: road segments connecting those points
• Weights on edges: estimated travel time, which can change with traffic

The app then runs an algorithm (often a variant of Dijkstra’s algorithm or A* search) to find the path with the smallest total weight from your starting vertex to your destination vertex.

This is one of the cleanest examples of introduction to graph theory with practical examples because you can literally see the graph on your screen as a map, and the app is constantly updating edge weights based on real-time data.

If you’re curious about how transportation networks are studied more formally, agencies like the U.S. Department of Transportation share data and research on network modeling and optimization at https://www.transportation.gov.

Social networks: who’s connected to whom (and how strongly)

Another modern example of graph theory you interact with constantly is a social network.

• Vertices: users
• Edges: friendships, follows, or connections
• Edge direction: on some platforms, follows are one-way (directed); on others, friendships are mutual (undirected)
• Weights: how often two people interact, or how “strong” a tie is

Social networks use graph theory to:

• Recommend friends based on mutual connections
• Detect communities or clusters of users with similar interests
• Identify influential users with many important connections

These are all real examples of graph theory in action. If you’ve ever seen a “People You May Know” suggestion, you’re seeing graph-based recommendation systems doing their thing behind the scenes.

Universities like MIT and Stanford publish accessible research and course notes on network science and graph theory; for instance, you can explore graph-related teaching material via MIT OpenCourseWare at https://ocw.mit.edu.

Internet and web links: how search engines rank pages

The web itself is a gigantic graph.

• Vertices: websites or individual web pages
• Edges: hyperlinks from one page to another (directed edges)

Search engines use graph theory to measure which pages are more “important” or “authoritative.” Google’s original PageRank algorithm is a famous example of graph theory applied to ranking.

In that setting:

• A page gets more “credit” if many other pages link to it
• Links from already-important pages count even more

This is a powerful example of introduction to graph theory with practical examples because it shows how a simple idea—counting and weighting links in a graph—can power a massive, global system.

For more background on how networks and algorithms are studied academically, you can look at resources like Harvard’s CS courses and reading lists at https://cs.harvard.edu.

Logistics, delivery routes, and the traveling salesperson

Imagine you run a small delivery business. You have a list of customers to visit and you want to:

• Start from your warehouse
• Visit every customer once
• Return to the warehouse
• Use the shortest possible route

This is the classic Traveling Salesperson Problem (TSP).

• Vertices: warehouse and customer locations
• Edges: possible roads between locations
• Weights: distance or time between locations

Finding the perfect route is hard when the number of locations grows large—this is a famous hard problem in computer science. But there are good approximation methods, and graph theory is the language we use to describe and attack it.

Delivery companies, ride-sharing apps, and even food delivery platforms use variants of this idea. These are some of the best examples of how graph theory translates directly into saving money, fuel, and time in the real world.

Operations research groups and transportation researchers, including many funded by the National Science Foundation (NSF), routinely publish papers on routing and optimization that are deeply rooted in graph theory. You can explore some of this at https://www.nsf.gov.

Power grids, communication networks, and reliability

Graph theory isn’t just about finding paths; it’s also about keeping things connected when something breaks.

Think of a power grid:

• Vertices: power plants, substations, and major consumers
• Edges: transmission lines

Engineers use graph theory to:

• Study how many lines or stations can fail before large areas lose power
• Design networks with redundancy (multiple paths) so a single failure doesn’t cause a blackout

Similar ideas apply to:

• Computer networks (routers and cables)
• Airline networks (airports and flight routes)

These real examples show how graph theory helps answer questions like, “Where is this network vulnerable?” and “Which nodes are so important that losing them would disconnect everything?”

Government agencies and research labs often publish work on infrastructure resilience using network models; the U.S. Department of Energy and national labs discuss grid reliability and network modeling at https://www.energy.gov.

Health, disease spread, and contact networks

Here’s a timely example of introduction to graph theory with practical examples: modeling how diseases spread.

Public health researchers often use contact networks:

• Vertices: people
• Edges: contacts close enough to transmit an infection

Graph theory helps them study:

• How quickly a disease might spread through a population
• Which individuals or locations act as super-spreaders
• What happens if you vaccinate or isolate certain nodes

During COVID-19, many models used network ideas to understand and predict spread. While the full models are complex, at the heart of many of them is a graph.

If you’re interested in the public health side, organizations like the CDC share educational material and research on disease modeling and networks at https://www.cdc.gov.

Basic graph theory concepts hiding in those examples

By now we’ve seen several examples of introduction to graph theory with practical examples: maps, social networks, web links, delivery routes, power grids, and disease spread. Let’s connect those to a few core concepts you’ll see in textbooks and problem sets.

Degree of a vertex

The degree of a vertex is the number of edges attached to it.

• In a social network, a person with many friends has high degree
• In an airline network, a big hub airport (like Atlanta or Chicago) has high degree

High-degree vertices can be very important: removing them might break the network into pieces.

Paths and cycles

A path is a sequence of vertices where each consecutive pair is connected by an edge.

• Your driving route from home to work is a path in the road graph

A cycle is a path that starts and ends at the same vertex without repeating edges.

• A delivery loop that starts and ends at the warehouse and visits several customers is a cycle

The traveling salesperson problem is about finding a special kind of cycle that visits every vertex once.

Connectedness

A graph is connected if you can get from any vertex to any other by some path.

• A disconnected road network would mean some towns have no route to others
• In a disconnected social graph, there are groups of people completely isolated from each other

In reliability and infrastructure planning, one of the main goals is to keep graphs connected even under failures.

Trees

A tree is a graph that is connected and has no cycles.

• A simple family tree is literally a tree
• Many computer data structures are trees (like file system directories)

Trees are especially nice to work with in problem solving, because they have predictable properties and no “loops” to complicate things.

These concepts show up repeatedly in the best examples of graph theory applications, from designing networks to analyzing data.

How graph theory shows up in 2024–2025 trends

Graph theory isn’t stuck in old textbooks; it’s right in the middle of current tech and research trends.

1. Graph neural networks (GNNs)

In modern machine learning, especially after 2020, graph neural networks have become a big deal. They’re used for:

• Recommender systems ("You might also like…")
• Fraud detection in financial transaction networks
• Analyzing molecular structures in drug discovery

In all of these, data is naturally graph-shaped, and GNNs are designed to learn from vertices and edges instead of rows and columns.

2. Supply chain resilience

Recent years have highlighted how fragile global supply chains can be. Companies and researchers now model supply chains as graphs:

• Vertices: factories, warehouses, ports, retailers
• Edges: shipping routes and transportation links

They use graph metrics to identify single points of failure and to design more resilient structures. These are timely real examples of graph theory directly affecting business and policy decisions.

3. Cybersecurity and attack graphs

Security teams model systems as graphs of computers, user accounts, and permissions. An attack graph shows possible paths an attacker could take through a network.

Graph algorithms help prioritize which vulnerabilities to fix first by identifying the most dangerous paths.

All of these modern topics are, in spirit, just more examples of introduction to graph theory with practical examples—the same basic ideas of vertices and edges, applied to new domains.

How to start solving graph theory problems yourself

If you’re ready to move from reading to doing, here’s a simple way to approach graph theory problems.

First, translate the story into a graph:

• Identify what the vertices should be (locations, people, web pages, etc.)
• Identify what the edges should be (roads, friendships, links, etc.)
• Decide if edges should have directions
• Decide if edges should have weights (distance, time, cost)

Next, match the question to a known graph problem:

• “Find the quickest route” usually means shortest path
• “Visit every location once and return” hints at traveling salesperson or Hamiltonian cycle
• “Can we connect everyone using the least total cost?” suggests a minimum spanning tree
• “Who is most influential?” often relates to centrality measures in networks

Finally, pick or learn an algorithm that matches the problem type. Even if you’re not coding, just recognizing the pattern puts you way ahead.

When you look back at all the examples of introduction to graph theory with practical examples in this article, you’ll notice they all follow that same pattern: identify vertices and edges, choose weights and directions, then ask a structured question about paths, connectivity, or importance.

FAQ: common questions about graph theory and real examples

Q1. What are some simple examples of graph theory in everyday life?
Subway maps, road navigation apps, airline routes, friendship networks, and delivery routes are all simple examples of graph theory. In each case, you have things (stations, cities, people) and connections (tracks, roads, flights, friendships) that can be modeled as a graph.

Q2. Can you give an example of a weighted graph I might recognize?
A city road map where each road segment has a travel time or distance is a perfect example of a weighted graph. Navigation apps use these weights to find the fastest or shortest path between two locations.

Q3. Are social networks directed or undirected graphs?
It depends on the platform. A platform with mutual friendships (both people must accept) is usually modeled as an undirected graph. A platform where you can follow someone without them following you back is better modeled as a directed graph. Both are common real examples in graph theory.

Q4. How is graph theory used in modern AI?
Graph theory underpins graph neural networks (GNNs), which are used for recommendation systems, analyzing molecules, fraud detection, and more. These models treat data as vertices and edges instead of just rows and columns, which makes them powerful for modern applications.

Q5. Where can I learn more with real, math-focused examples?
University courses and open materials are great starting points. For instance, MIT OpenCourseWare and Harvard’s computer science pages provide lecture notes and problem sets that include many examples of graph theory problems and solutions.

If you remember nothing else, remember this: whenever you can describe a situation as “things and connections between those things,” you’re probably looking at a graph. And once you see it that way, you can borrow all the power of graph theory—and all the examples of introduction to graph theory with practical examples you’ve just seen—to analyze and solve the problem.