---
tag: BDZGK4PHPR
kind: statement
role: theorem
name: Kirchhoff's theorem for multigraphs
sources:
  - wikipedia: en:1270691@r1374532193
    locator: Kirchhoff's theorem, section Particular cases and generalizations > Kirchhoff's theorem for multigraphs
    note: adopted
    mention: wikipedia://en/1270691#D0O8
    sections: [Statement]
inventory: [T7ZSPAQCV3]
---

## Statement

Kirchhoff's theorem holds for [[multigraphs|0C62PZXX02]] as well; the matrix *L* is modified as follows:

- The entry *q<sub>i,j</sub>* equals −*m*, where *m* is the number of edges between *i* and *j*;
- when counting the degree of a vertex, all loops are excluded.

Cayley's formula for a complete multigraph is *m*<sup>*n*−1</sup>(*n*<sup>*n*−1</sup>−(*n*−1)*n*<sup>*n*−2</sup>) by same methods produced above, since a simple graph is a multigraph with *m* = 1.