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Closed walks of length $n$ in $K_{p}$ are counted by $(p - 1)^{n} + (p - 1) (- 1)^{n}$

Example

For integers $p, n \geq 1$ , the number of rooted closed walks of length $n$ in the complete graph $K_{p}$ is

$(p - 1)^{n} + (p - 1) (- 1)^{n} .$

Here an undirected edge is represented by one directed edge in each direction, every edge has weight $1 \in Q$ , and a starting vertex is part of a rooted closed walk.

Facts & Assumptions

Given: Integers $p, n \geq 1$ and the complete graph $K_{p}$ with transfer matrix $A = J - I$ over $Q$ .

[L1]

A finite unit-weighted directed graph has a transfer matrix whose rows are sources and columns are targets (Finite weighted directed multigraphs, weighted walks and their transfer matrices).

[L2]

The diagonal entry $(A^{n})_{u u}$ counts length- $n$ walks from $u$ back to $u$ , so their total number is $tr (A^{n})$ (The $(u, v)$ entry of $A^{n}$ is the total weight of length- $n$ walks from $u$ to $v$ ).

[L3]

If $χ_{A} (t) = \prod_{i < p} (t - λ_{i})$ over the base field, then $tr (A^{n}) = \sum_{i < p} λ_{i}^{n}$ (Let $K$ be a field, $p \geq 1$ , and $A \in M_{p} (K)$ . If $χ_{A} (t) = \prod_{i < p} (t - λ_{i})$ in $K [t]$ , then the transfer-matrix trace series is $\sum_{i < p} (1 - λ_{i} x)^{- 1}$ ).

Verification

technique · explicit eigenbasis

1.1givenL1algebra

By [L1], $A = J - I$ . The all-ones vector $1$ satisfies $A 1 = (p - 1) 1$ , while $A (e_{i} - e_{0}) = - (e_{i} - e_{0})$ for $1 \leq i < p$ .

1.2givenalgebra

The list $1, e_{1} - e_{0}, \dots, e_{p - 1} - e_{0}$ is linearly independent over $Q$ : a relation has coordinates $α + β_{i} = 0$ for $i \geq 1$ and $α - \sum_{i} β_{i} = 0$ , hence $p α = 0$ . Since $p \neq 0$ in $Q$ , every coefficient is zero. It is therefore a basis of $Q^{p}$ .

2.1step 1.1step 1.2algebra

In the basis of step 1.2, $A$ is diagonal with entries $p - 1, - 1, \dots, - 1$ , so $χ_{A} (t) = (t - (p - 1)) (t + 1)^{p - 1}$ in $Q [t]$ .

3.1step 2.1L2L3∎

Apply [L3] to step 2.1 and then [L2] to obtain the displayed closed-walk count. When $p = 1$ , the difference-vector list is empty and the formula gives $0$ because $n \geq 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Example 4.7.5 (standard reference, not scraped)

Closed walks of length nn in KpKp​ are counted by (p−1)n+(p−1)(−1)n(p−1)n+(p−1)(−1)n