Incidence Matrix Calculator

Vertex × Edge matrix

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About Incidence Matrix Calculator

An incidence matrix calculator computing B(G): v×e matrix where B[i][j]=1 if vertex i is endpoint of edge j. B^T·B = L + 2I for simple graphs. Useful for network flow, cycle space, and cut space. Client-side.

Incidence Matrix Calculator Features

  • B matrix
  • Signed version
  • B^T·B relation
  • Cycle/cut space
  • Rank
Incidence matrix B: v×e matrix. B[i][j]=1 if vertex i is an endpoint of edge j. For signed version: orient edges, B[i][j]=+1 (tail), -1 (head). Then B·B^T = L (Laplacian). rank(B) = n - components.

How to Use

Enter edges:

  • B: Incidence matrix
  • Signed: Oriented version
  • Rank: n-components

Laplacian Connection

For the signed incidence matrix B̃ (orient each edge arbitrarily): B̃·B̃^T = L = D-A. This is independent of orientation! The null space of B̃^T is the cycle space. The row space is the cut space.

Applications

Network flow: conservation constraints. Electrical networks: Kirchhoff's laws. Algebraic topology: boundary operator. The incidence matrix is the fundamental object connecting graph theory and linear algebra.

Step-by-Step Instructions

  1. 1Enter edges.
  2. 2Build B matrix.
  3. 3Compute rank.
  4. 4Check B·B^T.
  5. 5Identify cycles.

Incidence Matrix Calculator — Frequently Asked Questions

What's the difference from adjacency matrix?+

Adjacency A is v×v (vertex-vertex). Incidence B is v×e (vertex-edge). A encodes which vertices are adjacent. B encodes which vertices belong to which edges. They contain the same information but in different formats.

What determines the rank?+

rank(B) = n - c where c = number of connected components. The null space has dimension e - n + c = dimension of cycle space (this is the circuit rank or cyclomatic number). These are fundamental graph invariants.

What about directed graphs?+

For digraphs: B[i][j]=+1 if edge j leaves vertex i, -1 if it enters. Then B·1 = 0 (each column sums to 0). The directed incidence matrix encodes flow conservation constraints.

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