Kirchhoff's theorem

Key takeaways

Kirchhoff's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kirchhoff's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kirchhoff's theorem from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem is a theorem about the number of spanning trees in a graph. It states that this number can be computed as any cofactor of the graph's Laplacian matrix. This shows in particular that the number of spanning trees can be computed from the graph data in polynomial time. Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. The theorem is named after the German mathematician Gustav Kirchhoff, who published it in 1847. An English translation of Kirchhoff's paper was published in 1958.

Definitions and statement Let G be a simple, undirected graph. A spanning tree of G is a subgraph of G that is a tree with the same vertex set as G. The Laplacian matrix L of G is the difference between the graph's degree matrix (the diagonal matrix of vertex degrees) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise). A cofactor of L is obtained by deleting a row (say the i-the row) and a column (say the j-th column) of L, taking the determinant of that smaller matrix, and multiplying it with (-1)i+j. Kirchhoff's theorem states that the number of spanning trees of the graph is equal to any cofactor of the graph's Laplacian matrix. (In particular, all these cofactors are equal.)

Example

First, construct the Laplacian matrix L for the example diamond graph G (see image on the right):

L = [ 2 − 1 − 1 0 − 1 3 − 1 − 1 − 1 − 1 3 − 1 0 − 1 − 1 2 ] . {\displaystyle L=\left[{\begin{array}{rrrr}2&-1&-1&0\\-1&3&-1&-1\\-1&-1&3&-1\\0&-1&-1&2\end{array}}\right].}

Next, construct a matrix Q* by deleting any row and any column from Q. For example, deleting row 1 and column 1 yields

L ∗ = [ 3 − 1 − 1 − 1 3 − 1 − 1 − 1 2 ] . {\displaystyle L^{\ast }=\left[{\begin{array}{rrr}3&-1&-1\\-1&3&-1\\-1&-1&2\end{array}}\right].}

Finally, take the determinant of L*, which results in 8. The number of spanning trees of G, shown on the right, is also 8.

Proof outline (The proof below is based on the Cauchy–Binet formula. An elementary induction argument for Kirchhoff's theorem can be found on page 654 of Moore (2011).) First notice that the Laplacian matrix has the property that the sum of its entries across any row and any column is 0. Thus we can transform any minor into any other minor by adding rows and columns, switching them, and multiplying a row or a column by −1. Thus the cofactors are the same up to sign, and it can be verified that, in fact, they have the same sign. We proceed to show that the determinant of the minor M11 is the number of spanning trees. Let n be the number of vertices of the graph, and m the number of its edges. The incidence matrix E is an n-by-m matrix, which may be defined as follows: suppose that (i, j) is the kth edge of the graph, and that i < j. Then Eik = 1, Ejk = −1, and all other entries in column k are 0 (see oriented incidence matrix for understanding this modified incidence matrix E). For the preceding example (with n = 4 and m = 5):

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Kirchhoff's theorem

Start with the simplest possible case. Write down what Kirchhoff's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kirchhoff's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kirchhoff's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kirchhoff's theorem

In research: Kirchhoff's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kirchhoff's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kirchhoff's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic graph theory, Gustav Kirchhoff, Spanning tree, so understanding it makes those chapters shorter.
In everyday life: Look for Kirchhoff's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Kirchhoff's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kirchhoff's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kirchhoff's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kirchhoff's theorem in simple terms?

In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem is a theorem about the number of spanning trees in a graph. It states that this number can be computed as any cofactor of the graph's Laplacian matrix.

Why does Kirchhoff's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kirchhoff's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kirchhoff's theorem.