Introduction to Graphs

Learn the fundamental vocabulary of graphs — vertices, edges, directed vs undirected, weighted, cyclic vs acyclic, and dense vs sparse — the building blocks for every graph algorithm.

4 min read Level 2/5 #dsa#graph#vertices

What you'll learn

Define graphs and distinguish directed from undirected graphs
Identify weighted, cyclic, acyclic, dense, and sparse graphs
Recognize real-world problems that map naturally onto graph structures

A graph is a collection of vertices (nodes) connected by edges. Unlike arrays or trees, graphs place no constraints on how nodes connect — any vertex can link to any other, making graphs the right tool for networks, maps, dependencies, and social relationships.

Vertices and Edges

A graph G is written G = (V, E) where V is the set of vertices and E is the set of edges. An edge connects two vertices. In JavaScript we usually represent a small example graph like this:

// 4 vertices: 0, 1, 2, 3
// Edges: 0-1, 0-2, 1-3, 2-3
const edges = [
  [0, 1],
  [0, 2],
  [1, 3],
  [2, 3],
];

Directed vs Undirected

In an undirected graph an edge between A and B means you can travel in either direction. In a directed graph (digraph) an edge A → B only lets you travel from A to B.

Real examples: a road network (undirected), Twitter follows (directed).

Weighted Graphs

A weighted graph assigns a numeric cost to each edge — distance, time, or capacity. Shortest-path algorithms like Dijkstra rely on these weights.

// Weighted edge list
const edges = [
  { from: 0, to: 1, weight: 4 },
  { from: 0, to: 2, weight: 1 },
  { from: 2, to: 1, weight: 2 },
];

Cyclic vs Acyclic

A graph is cyclic if there exists a path from some vertex back to itself. A graph with no such path is acyclic. A directed acyclic graph (DAG) is the backbone of topological sorting, build systems, and dependency resolution.

Dense vs Sparse

The number of edges relative to vertices determines density:

Term	Edge count	Typical representation
Sparse	Close to V	Adjacency list
Dense	Close to V²	Adjacency matrix

Most real-world graphs (social networks, road maps) are sparse, which is why adjacency lists are the default choice in JavaScript.

Up Next

Now that you know the vocabulary, learn how to store a graph in memory and compare the tradeoffs of each representation.

Graph Representations →