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Kirchhoff's formula gives $τ (K_{4}) = 16$

Example

The complete graph $K_{4}$ has exactly $16$ spanning trees.

Facts & Assumptions

Given: The complete graph $K_{4}$ .

[L1]

For a regular graph, $τ (G) = \frac{1}{∣ V (G) ∣} \prod_{j = 2}^{∣ V (G) ∣} (d - λ_{j})$ (The matrix-tree theorem becomes an eigenvalue product formula).

[L2]

The adjacency eigenvalues of $K_{4}$ are $3, - 1, - 1, - 1$ (The complete graph $K_{n}$ has adjacency spectrum ${n - 1, (- 1)^{n - 1}}$ ).

Verification

technique · direct

1.1

The graph $K_{4}$ is $3$ -regular and has four vertices, so [L1] and [L2] give $τ (K_{4}) = \frac{1}{4} (3 - (- 1))^{3} = \frac{1}{4} \cdot 4^{3} = 16$ .

L1L2algebra

2.1

Therefore Kirchhoff's product formula recovers the count $τ (K_{4}) = 16$ .

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard P. Stanley, MIT 18.314 handout, Example 1.11 (standard reference, not scraped)

Kirchhoff's formula gives τ(K4)=16τ(K4​)=16