Adjacency Matrix Calculator

Convert between adjacency matrix, edge list, and adjacency list. Auto-detects directed/undirected graphs, computes degree sequence, density, connected components, and matrix powers — with an interactive SVG graph visualization.

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About Adjacency Matrix Calculator

The Adjacency Matrix Calculator is a graph-theory utility that converts between the three canonical graph representations — adjacency matrix, edge list, and adjacency list — and enriches the result with structural analysis: degree sequence, graph density, connected components, and matrix powers. It auto-detects whether your input describes a directed or undirected graph and renders a live SVG visualization alongside every result.

What Is an Adjacency Matrix?

Given a graph G = (V, E) with n vertices, its adjacency matrix is the n × n square matrix A whose entry A[i][j] is 1 if there is an edge from vertex i to vertex j, and 0 otherwise.

A[i][j] = 1 if (v_i, v_j) ∈ E , else 0

For an undirected graph, the adjacency matrix is always symmetric: every edge {u, v} contributes both A[u][v] = 1 and A[v][u] = 1. For a directed graph (digraph), the matrix may be asymmetric, reflecting the direction of each arc.

Three Representations — Pick What Fits Your Problem

Representation	Space	Edge lookup	List neighbors	Best for
Adjacency matrix	Θ(n²)	O(1)	Θ(n)	Dense graphs; matrix algebra (powers, eigenvalues)
Adjacency list	Θ(n + m)	O(deg v)	Θ(deg v)	Sparse graphs; BFS/DFS and shortest-path algorithms
Edge list	Θ(m)	Θ(m)	Θ(m)	Input/output, Kruskal's MST, edge-centric algorithms

Key Metrics Computed

Degree Sequence

For undirected graphs, the degree of a vertex is the number of edges incident to it (with self-loops counting twice). For directed graphs, each vertex has an in-degree (incoming arcs) and out-degree (outgoing arcs). The sorted list of degrees is a classical graph invariant used in isomorphism testing and the Erdős–Gallai realisability theorem.

Handshaking Lemma: Σ deg(v) = 2m (undirected) Σ in-deg(v) = Σ out-deg(v) = m (directed)

Graph Density

Density measures how "full" a graph is relative to the maximum number of edges possible on n vertices.

Undirected: D = 2m / (n(n−1)) Directed: D = m / (n(n−1))

A density of 0 means no edges, 1 means the graph is complete, and values below 0.1 typically indicate a sparse graph where an adjacency list is more space-efficient than a matrix.

Connected Components

A connected component is a maximal subset of vertices such that every pair is joined by a path. For directed graphs, this calculator reports weakly connected components (ignoring arrow directions) — the same subsets you'd get by treating each arc as an undirected edge.

Matrix Powers (A², A³ ... )

A fundamental theorem of algebraic graph theory states that the (i, j) entry of A^k equals the number of walks of exactly length k from vertex i to vertex j. Consequently:

A²[i][i] equals the degree of vertex i (undirected), since a 2-walk from i to itself is "go to a neighbor and back".
The trace of A³ divided by 6 counts the triangles in an undirected graph.
Whether Aⁿ⁻¹ has any zero entry tells you whether the graph is connected.

Input Formats Accepted

1. Edge list

One edge per line or comma-separated. Any of these separators work: A-B, A B, A,B, A->B, A--B. Use -> if you want to force a directed interpretation.

A-B, B-C, C-A, C-D (undirected 4-cycle with a tail) A->B, B->C, C->D, D->A (directed cycle of length 4)

2. Adjacency list

One line per vertex, in the form vertex: neighbor1, neighbor2, .... Order doesn't matter; missing vertices are added automatically from the neighbor lists.

A: B, C, D B: A, C C: A, B, D D: A, C

3. Adjacency matrix

One row per line with space- or comma-separated 0/1 values. The matrix must be square. Optionally provide custom labels in the Matrix labels field (otherwise A, B, C… are used).

0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 0

How to Use This Calculator

Pick an input format using the tabbed selector: edge list, adjacency list, or adjacency matrix.
Paste or type your graph in the text area. For matrix input, add optional labels in the Matrix labels field.
Select graph type — leave on Auto-detect and the calculator will infer directedness from arrows (->) or matrix symmetry. Force it to Directed or Undirected if you want to override.
Click Convert & Analyze Graph. The result page shows the adjacency matrix, an interactive SVG rendering, the other two text representations, degree statistics, connected components, and walk-count matrices A² and A³ when the graph is small enough.
Hover a matrix row or a graph node to light up the matching row/column and incident edges — an instant visual proof that each format encodes the same information.

Worked Example

Consider an undirected graph on vertices {A, B, C, D} with edges AB, BC, CA, CD. The adjacency matrix is:

A B C D A [ 0 1 1 0 ] B [ 1 0 1 0 ] C [ 1 1 0 1 ] D [ 0 0 1 0 ]

Key facts the calculator derives:

Symmetric? Yes — confirms undirected.
Degree sequence: (3, 2, 2, 1) — vertex C is the hub.
Density: 2·4 / (4·3) = 0.667 — a moderately dense graph.
Connected? Yes, single component.
Triangles: exactly one (A–B–C), as confirmed by tr(A³) = 6.

Common Applications

Social network analysis — friendship / follower graphs, centrality.
Web & citation graphs — PageRank and HITS work directly on A and A^T.
Routing & networks — shortest-path, min-cut, max-flow.
Chemistry — molecular graphs with atoms as vertices, bonds as edges.
Scheduling & dependency resolution — directed acyclic graphs (DAGs) in build systems.
Markov chains — row-stochastic matrices derived from graphs encode transition probabilities.

Frequently Asked Questions

What is an adjacency matrix?

An adjacency matrix is an n × n square matrix used to represent a finite graph. Each cell A[i][j] is 1 if there is an edge from vertex i to vertex j, and 0 otherwise. For undirected graphs the matrix is symmetric, so A[i][j] = A[j][i]. The matrix makes it easy to check whether two vertices are connected in constant time, and matrix powers encode the number of walks between vertices.

How do I tell if a graph is directed from its adjacency matrix?

If the adjacency matrix is symmetric, meaning A[i][j] equals A[j][i] for every pair of indices, the graph is undirected. If there is at least one pair where A[i][j] differs from A[j][i], the graph is directed. This calculator performs that symmetry check automatically when you pick the Auto-detect option.

What does the k-th power of an adjacency matrix represent?

The entry (i, j) of A^k counts the number of walks of exactly length k from vertex i to vertex j. For example, A²[i][j] is the number of 2-step paths, which equals the number of common neighbors between i and j in undirected graphs. This property is used in algorithms for triangle counting, reachability, and PageRank-style computations.

What is graph density?

Graph density is the ratio of the number of edges present to the maximum possible number of edges. For an undirected simple graph with n vertices, density = 2m / (n(n-1)). For a directed graph, density = m / (n(n-1)). A density near 0 means a sparse graph; a density of 1 means a complete graph.

How is an adjacency matrix different from an adjacency list?

An adjacency matrix stores connectivity for every pair of vertices using n² bits, making neighbor lookup O(1) but memory usage O(n²). An adjacency list stores only the actual neighbors of each vertex, giving O(n + m) memory, which is far smaller for sparse graphs, but neighbor lookup requires a linear scan. Matrices are better for dense graphs and matrix-algebra operations; lists are better for sparse graphs and traversal algorithms like BFS/DFS.

Can this tool handle weighted graphs?

The current calculator focuses on unweighted adjacency matrices with 0/1 entries. If you paste a matrix with non-zero numeric weights, every non-zero cell is treated as a 1 for structural analysis. For weighted graph computations such as shortest-path, consider a dedicated weighted-graph tool.