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Mathematics

Introduction to Graph Theory

Vertices, Edges, Euler Paths, and Network Theorems — A TLDR Primer

Discrete math courses move fast, and graph theory is usually the section where students fall behind first. The vocabulary is dense, the diagrams look deceptively simple, and by the time Euler paths and adjacency matrices show up on the exam, it feels like the foundation was never built properly. This guide builds that foundation.

**TLDR: Introduction to Graph Theory** covers everything a high school or early college student needs to get oriented: what a graph actually is (and why it has nothing to do with the xy-plane), how to represent graphs as pictures, matrices, and adjacency lists, and how to read the structural vocabulary — degree, paths, cycles, connectivity — that every later topic depends on. From there, the book works through the three graph families that appear everywhere (trees, bipartite graphs, and planar graphs), tackles the classic problems of Euler and Hamilton paths, and closes with a clear walkthrough of Dijkstra's shortest-path algorithm.

This is a focused primer for discrete math study, not an exhaustive textbook. Every section leads with the one idea you need to take away, backs it up with worked examples and real numbers, and flags the misconceptions that trip students up most often. If you're prepping for a discrete mathematics course, studying for a coding interview, or helping a student who's lost in a graph theory unit, this is the 15-page pass that makes the full course make sense.

Pick it up, read it in an afternoon, and walk in ready.

What you'll learn
  • Define graphs, vertices, edges, and the main types (directed, weighted, simple, multigraph).
  • Translate between visual graphs, adjacency matrices, and adjacency lists.
  • Use degree, paths, cycles, and connectivity to describe graph structure.
  • Identify trees, bipartite graphs, and planar graphs and state what makes them special.
  • Apply Euler's and Hamilton's path conditions and reason about shortest paths.
  • Recognize where graph theory shows up in computer science, biology, and everyday networks.
What's inside
  1. 1. What Is a Graph?
    Introduces graphs as collections of vertices and edges, distinguishes graph-theory graphs from xy-plane graphs, and sets up the basic vocabulary used throughout the book.
  2. 2. Representing Graphs: Pictures, Matrices, and Lists
    Shows three equivalent ways to write down a graph and when to use each, with worked conversions between them.
  3. 3. Degree, Paths, and Connectivity
    Develops the structural vocabulary — degree, walks, paths, cycles, components — and proves the handshake lemma.
  4. 4. Special Graphs: Trees, Bipartite Graphs, and Planar Graphs
    Introduces three families of graphs that show up everywhere and the simple rules that characterize them.
  5. 5. Classic Problems: Euler Paths, Hamilton Paths, and Shortest Paths
    Walks through the Königsberg bridges, Hamilton circuits, and the idea behind Dijkstra's shortest-path algorithm.
  6. 6. Why It Matters: Graphs in the Real World
    Connects graph theory to social networks, the internet, biology, and computer science to motivate further study.
Published by Solid State Press
Introduction to Graph Theory cover
TLDR STUDY GUIDES

Introduction to Graph Theory

Vertices, Edges, Euler Paths, and Network Theorems — A TLDR Primer
Solid State Press

Introduction to Graph Theory

Vertices, Edges, Euler Paths, and Network Theorems — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Is a Graph?
  2. 2 Representing Graphs: Pictures, Matrices, and Lists
  3. 3 Degree, Paths, and Connectivity
  4. 4 Special Graphs: Trees, Bipartite Graphs, and Planar Graphs
  5. 5 Classic Problems: Euler Paths, Hamilton Paths, and Shortest Paths
  6. 6 Why It Matters: Graphs in the Real World
Chapter 1

What Is a Graph?

Forget the xy-plane for a moment. In graph theory, a graph is something much simpler: a collection of objects and the connections between them. That's it. The objects are called vertices (singular: vertex), and the connections are called edges. You can think of vertices as dots and edges as lines drawn between them.

Formally, a graph $G$ is written as $G = (V, E)$, where $V$ is the set of vertices and $E$ is the set of edges. Each edge connects a pair of vertices, called its endpoints. If an edge connects vertex $u$ to vertex $v$, we write it as the pair $\{u, v\}$.

Example. Suppose you have four cities — Albany, Boston, Chicago, and Denver — and direct flights between Albany–Boston, Albany–Chicago, and Boston–Denver. Describe this as a graph.

Solution. The vertex set is $V = \{\text{Albany, Boston, Chicago, Denver}\}$. The edge set is $E = \{\{\text{Albany, Boston}\},\ \{\text{Albany, Chicago}\},\ \{\text{Boston, Denver}\}\}$. Drawn on paper, you'd place four dots (one per city) and three line segments connecting the appropriate pairs.

One thing to clear up immediately: this kind of graph has nothing to do with plotting $y = x^2$ on coordinate axes. A common mistake is to confuse the two uses of the word "graph." In graph theory, there are no axes, no functions, and no coordinates. A graph theory graph is a structural diagram showing relationships, not a picture of a function. From here on, "graph" always means the graph-theory kind.

Directed and Undirected Graphs

In the city example above, the edge between Albany and Boston represents a two-way connection — you can fly either direction. That makes it an undirected graph: edges have no built-in direction. The edge $\{u, v\}$ is the same as $\{v, u\}$.

About This Book

If you are a high school student encountering graph theory for the first time, or a college freshman working through a discrete mathematics course and hitting a wall on vertices and edges, this book is for you. It also works for anyone using graphs as coding interview math prep — the kind of graph and network problems that show up in technical screens at every major tech company.

This primer covers the core ideas you need: what a graph is, how to represent one as a matrix or adjacency list, how to measure degree and connectivity, and how trees, bipartite graphs, and planar graphs differ. It explains Euler paths, Hamilton paths, and Dijkstra's shortest-path algorithm. A concise overview with no filler.

Read straight through to build the framework, then work every example inline. The practice problems with solutions at the end let you check whether the ideas have actually stuck.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

Coming soon to Amazon