794 lines
24 KiB
Plaintext
794 lines
24 KiB
Plaintext
---
|
|
title: "Generating Polynomials, Part 2: Ghostly Chains"
|
|
description: |
|
|
"What do polygons without distance still know about planar geometry?"
|
|
format:
|
|
html:
|
|
html-math-method: katex
|
|
date: "2021-08-19"
|
|
date-modified: "2025-06-24"
|
|
categories:
|
|
- algebra
|
|
- linear algebra
|
|
- generating functions
|
|
- graph theory
|
|
- python
|
|
---
|
|
|
|
<style>
|
|
.figure-img {
|
|
max-width: 512px;
|
|
object-fit: contain;
|
|
height: 100%;
|
|
}
|
|
|
|
.figure-img.wide {
|
|
max-width: 768px;
|
|
}
|
|
</style>
|
|
|
|
In the [previous post](../1), I tied the geometry regular polygons to a sequence of polynomials
|
|
though some clever algebraic manipulation.
|
|
But let's deign to ask a very basic question: what is a polygon?
|
|
|
|
|
|
Loops without Distance
|
|
----------------------
|
|
|
|
Fundamentally, a polygon is just a collection of vertices and edges.
|
|
For polygons in a Euclidean setting, the position of points matters,
|
|
as well as the lines connecting them -- a rectangle is different from a trapezoid or a kite.
|
|
But at its simplest, this is just a tabulation of points and adjacencies.
|
|
|
|
|
|
|
|
Only examining these figures by their connectedness is precisely the kind of thing
|
|
*graph theory* deals with.
|
|
"Graph" is a potentially confusing term, since it has nothing to do with "graphs of functions",
|
|
but the name is supposed to evoke the fact that they are "drawings".
|
|
For the graphs we're interested in, there's some additional terminology:
|
|
|
|
- Vertices themselves are sometimes instead called *nodes*
|
|
- Edge in the graph have no direction in how they connect nodes, so the graph is called *undirected*.
|
|
- If the nodes in a graph can be arranged so that no edges appear to intersect,
|
|
the graph is *planar*.
|
|
- For example, lower-right figure in the above diagram appears to have intersecting edges,
|
|
but the nodes can be rearranged to look like the other graphs, so it is planar.
|
|
|
|
It's easiest to study families of graphs, rather than isolated examples.
|
|
If the graph is a simple loop of nodes, it is called a
|
|
[*cycle graph*](https://en.wikipedia.org/wiki/Cycle_graph).
|
|
They are denoted by $C_n$, where *n* is the number of nodes.
|
|
In a cycle graph, since all nodes are identical to each other (they all connect to two edges)
|
|
and all edges are identical to each other (they connect identical vertices),
|
|
the best geometric interpretation is a shape which is
|
|
|
|
- Regular, so that each edge and each angle (vertex) are of equal measure
|
|
- Convex, so that no edge meets another without creating a vertex (or node)
|
|
|
|
In other words, $C_3$ is analogous to an equilateral triangle, $C_4$ is analogous to a square, and so on.
|
|
|
|
|
|
### Encoding Graphs
|
|
|
|
There are two primary ways to store information about a graph.
|
|
The first is by labelling each node (for example, with integers), then recording the edges as
|
|
a list of pairs of connected nodes.
|
|
In the case of an undirected graph, these are unordered pairs.
|
|
While such a list is convenient, it doesn't convey a lot of information about the graph
|
|
besides the number of edges.
|
|
|
|
Alternatively, these pairs can also be interpreted as addresses in a square matrix,
|
|
called an *adjacency matrix*.
|
|
Each column and row correspond to a specific node, and an entry is 1
|
|
when the nodes of a row and column of are joined by an edge (and 0 otherwise).
|
|
For undirected graphs, these matrices are symmetric, since it is possible
|
|
to traverse an edge in either direction.
|
|
|
|
$$
|
|
\begin{align*}
|
|
C_3 := \begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 0)
|
|
]\end{matrix} & \cong
|
|
\begin{pmatrix}
|
|
0 & 1 & 1 \\
|
|
1 & 0 & 1 \\
|
|
1 & 1 & 0
|
|
\end{pmatrix}
|
|
\\ \\
|
|
C_4 := \begin{matrix} [
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 3), \\
|
|
(3, 0)
|
|
]\end{matrix} & \cong
|
|
\begin{pmatrix}
|
|
0 & 1 & 0 & 1 \\
|
|
1 & 0 & 1 & 0 \\
|
|
0 & 1 & 0 & 1 \\
|
|
1 & 0 & 1 & 0
|
|
\end{pmatrix}
|
|
\\ \\
|
|
C_5 := \begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 3), \\
|
|
(3, 4), \\
|
|
(4, 0)
|
|
]\end{matrix} &\cong
|
|
\begin{pmatrix}
|
|
0 & 1 & 0 & 0 & 1 \\
|
|
1 & 0 & 1 & 0 & 0 \\
|
|
0 & 1 & 0 & 1 & 0 \\
|
|
0 & 0 & 1 & 0 & 1 \\
|
|
1 & 0 & 0 & 1 & 0
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
$$
|
|
|
|
Swapping the labels on two nodes is will exchange two rows and two columns
|
|
of the adjacency matrix.
|
|
Just one of these swaps would flip the sign of its determinant, but since they occur in pairs,
|
|
the determinant is invariant of the labelling (equally, a graph invariant).
|
|
|
|
|
|
Prismatic Recurrence
|
|
--------------------
|
|
|
|
The determinant of a matrix is also the product of its eigenvalues, which are another matrix invariant.
|
|
The set of eigenvalues is also called its *spectrum*, and the study of the spectra of graphs is called
|
|
[*spectral graph theory*](https://en.wikipedia.org/wiki/Spectral_graph_theory)[^1],
|
|
|
|
[^1]: It is also among the most mystifying names in math to read without any context
|
|
|
|
Eigenvalues are the roots of the characteristic polynomial of a matrix.
|
|
The matrix $C_5$ is sufficiently large enough to generalize to $C_n$, and its characteristic polynomial by
|
|
[Laplace expansion](https://en.wikipedia.org/wiki/Laplace_expansion) is:
|
|
|
|
$$
|
|
\begin{gather*}
|
|
Ax = \lambda x \implies (\lambda I - A)x = 0
|
|
\\ \\
|
|
c_5(\lambda) = |\lambda I - C_5|
|
|
= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 & -1 \\
|
|
-1 & \lambda & -1 & 0 & 0 \\
|
|
0 & -1 & \lambda & -1 & 0 \\
|
|
0 & 0 & -1 & \lambda & -1 \\
|
|
-1 & 0 & 0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
= \lambda m_{1,1}
|
|
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 2 \ }}^\text{sign} m_{1, 2}
|
|
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 5 \ }}^\text{sign} m_{1, 5}
|
|
\end{gather*}
|
|
$$
|
|
|
|
Note that every occurrence of "5" generalizes to higher *n*.
|
|
The first [minor](https://en.wikipedia.org/wiki/Matrix_minor)
|
|
is easily expressed in terms of *another* matrix's characteristic polynomial.
|
|
|
|
$$
|
|
m_{1, 1}[C_5]
|
|
= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 \\
|
|
-1 & \lambda & -1 & 0 \\
|
|
0 & -1 & \lambda & -1 \\
|
|
0 & 0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
= |\lambda I - P_4| = p_{5 - 1}(\lambda)
|
|
$$
|
|
|
|
We will come to the meaning of the $P_n$ in a moment.
|
|
The other minors require extra expansions, but ones that (thankfully) quickly terminate.
|
|
|
|
$$
|
|
\begin{matrix}
|
|
m_{1, 2}[C_5]
|
|
&= \left |
|
|
\begin{matrix}
|
|
-1 & -1 & 0 & 0 \\
|
|
0 & \lambda & -1 & 0 \\
|
|
0 & -1 & \lambda & -1 \\
|
|
-1 & 0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
&=& (-1)
|
|
\left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 \\
|
|
-1 & \lambda & -1 \\
|
|
0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
&+& (-1)(-1)^{1 + 4}
|
|
\left |
|
|
\begin{matrix}
|
|
-1 & 0 & 0 \\
|
|
\lambda & -1 & 0 \\
|
|
-1 & \lambda & -1
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&&=& (-1)|\lambda I - P_3|
|
|
&+& (-1)\overbrace{(-1)^{5}(-1)^{5 - 2}}^{\text{even power, even when $\scriptsize n \neq 5$}}
|
|
\\
|
|
&&=& (-1)p_{5 - 2}(\lambda) &+& (-1)
|
|
\\
|
|
&&=& -(p_{5 - 2}(\lambda) &+& 1)
|
|
\end{matrix}
|
|
$$
|
|
|
|
The "1 + 4" exponent when evaluating this minor comes from the address of the lower-left -1, (i.e., (1, 4)).
|
|
This entry exists for all $C_n$.
|
|
The determinant of the rightmost matrix is just the product of the -1's on the diagonal, so it will always
|
|
have a power of the same parity as *n*, which cancels out with the sign of the minor.
|
|
Meanwhile, another $P$-type matrix appears in the other term, this time of two lower orders.
|
|
|
|
$$
|
|
\begin{matrix}
|
|
\\ \\
|
|
m_{1, 5}[C_5] &=
|
|
\left |
|
|
\begin{matrix}
|
|
-1 & \lambda & -1 & 0 \\
|
|
0 & -1 & \lambda & -1 \\
|
|
0 & 0 & -1 & \lambda \\
|
|
-1 & 0 & 0 & -1
|
|
\end{matrix}
|
|
\right |
|
|
&=& (-1)
|
|
\left |
|
|
\begin{matrix}
|
|
-1 & \lambda & -1 \\
|
|
0 & -1 & \lambda \\
|
|
0 & 0 & -1
|
|
\end{matrix}
|
|
\right |
|
|
&+& (-1)(-1)^{5 - 2}
|
|
\left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 \\
|
|
-1 & \lambda & -1 \\
|
|
0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&&=& (-1)(-1)^{5-2} &+& (-1)(-1)^{5 - 2}|\lambda I - P_3|
|
|
\\
|
|
&&=& (-1)^{5-1}((-1)(-1) &+& (-1)(-1)p_{5 - 2}(\lambda))
|
|
\\
|
|
&&=& (-1)^{5-1}(1 &+& p_{5 - 2}(\lambda))
|
|
\end{matrix}
|
|
$$
|
|
|
|
A third $P$-type matrix appears, just like the other minor.
|
|
Unfortunately, this minor *does* depend on the parity of *n*.
|
|
|
|
All together, this produces a characteristic polynomial in terms of the polynomials $p_n(\lambda)$:
|
|
|
|
$$
|
|
\begin{align*}
|
|
&& c_5(\lambda) &= \lambda p_{5 - 1}
|
|
+ (-1)(p_{5 - 2} + 1)
|
|
+ (-1)\overbrace{(-1)^{5 - 1} (-1)^{5 - 1}}^{\text{even, even when $\scriptsize n \neq 5$}}(p_{5 - 2} + 1)
|
|
\\
|
|
&&&= \lambda p_{5 - 1}
|
|
- (p_{5 - 2} + 1)
|
|
- (p_{5 - 2} + 1)
|
|
\\
|
|
&&&= \lambda p_{5 - 1}
|
|
- 2(p_{5 - 2} + 1)
|
|
\\
|
|
&& \implies c_n(\lambda) &= \lambda p_{n - 1}(\lambda)
|
|
- 2(p_{n - 2}(\lambda) + 1)
|
|
\end{align*}
|
|
$$
|
|
|
|
Fortunately, the minor whose determinant depended on the parity of *n* cancels with $(-1)^{1 + 5}$,
|
|
and the resulting expression seems to generically apply across all *n*.
|
|
Further, this resembles a recurrence relation, which is great for building a rule.
|
|
But it is meaningless without knowing $p_n(\lambda)$.
|
|
|
|
|
|
Powerful Chains
|
|
---------------
|
|
|
|
The various $P_n$ are in fact the adjacency matrices of a path on *n* nodes.
|
|
|
|
{.wide}
|
|
|
|
$$
|
|
\begin{align*}
|
|
P_2 &:=
|
|
\begin{matrix}[
|
|
(0, 1)
|
|
]\end{matrix}
|
|
\cong \begin{pmatrix}
|
|
0 & 1 \\
|
|
1 & 0
|
|
\end{pmatrix}
|
|
\\
|
|
P_3 &:=
|
|
\begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2)
|
|
]\end{matrix}
|
|
\cong \begin{pmatrix}
|
|
0 & 1 & 0 \\
|
|
1 & 0 & 1 \\
|
|
0 & 1 & 0
|
|
\end{pmatrix}
|
|
\\
|
|
P_4 &:=
|
|
\begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 3)
|
|
]\end{matrix}
|
|
\cong \begin{pmatrix}
|
|
0 & 1 & 0 & 0 \\
|
|
1 & 0 & 1 & 0 \\
|
|
0 & 1 & 0 & 1 \\
|
|
0 & 0 & 1 & 0
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
$$
|
|
|
|
These matrices are similar to the ones for cycle graphs, but lack the entries in bottom-left
|
|
and upper-right corners.
|
|
Consequently, the characteristic polynomials of $P_n$ are much easier to solve for.
|
|
|
|
$$
|
|
\begin{gather*}
|
|
p_4(\lambda) = |\lambda I - P_4|
|
|
= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 \\
|
|
-1 & \lambda & -1 & 0 \\
|
|
0 & -1 & \lambda & -1 \\
|
|
0 & 0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\ \\
|
|
= \lambda \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 \\
|
|
-1 & \lambda & -1 \\
|
|
0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right | + (-1)(-1)^{1+2} \left |
|
|
\begin{matrix}
|
|
-1 & -1 & 0 \\
|
|
0 & \lambda & -1 \\
|
|
0 & -1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\ \\
|
|
= \lambda |\lambda I - P_3| + \left (
|
|
(-1) \left |
|
|
\begin{matrix}
|
|
\lambda & -1 \\
|
|
-1 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
+ (-1)(-1) \left |
|
|
\begin{matrix}
|
|
0 & -1 \\
|
|
0 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\right)
|
|
\\ \\
|
|
= \lambda |\lambda I - P_3| - |\lambda I - P_2|
|
|
\\
|
|
= \lambda p_{4 - 1}(\lambda) - p_{4 - 2}(\lambda)
|
|
\\
|
|
\implies p_{n}(\lambda) = \lambda p_{n - 1}(\lambda) - p_{n - 2}(\lambda)
|
|
\end{gather*}
|
|
$$
|
|
|
|
While the earlier equation for $c_n$ in terms of $p_n$ reminded of a recurrence relation,
|
|
*this* actually is one (and it should look familiar).
|
|
|
|
Since the recurrence has order 2, it requires two initial terms: $p_0$ and $p_1$.
|
|
The graph corresponding to $p_1$ is a single node, not connected to anything.
|
|
Therefore, its adjacency matrix is a 1x1 matrix with 0 as its only entry,
|
|
and its characteristic polynomial is $\lambda$.
|
|
By the recurrence, $p_2 = \lambda p_1 -\ p_0 = \lambda^2 -\ p_0$.
|
|
Equating terms with the characteristic polynomial of $P_2$, it is obvious that
|
|
|
|
$$
|
|
|\lambda I - P_2|
|
|
= \begin{pmatrix}
|
|
\lambda & -1 \\
|
|
-1 & \lambda
|
|
\end{pmatrix}
|
|
= \lambda^2 - 1 = \lambda p_1 - p_0 \\
|
|
\implies p_0 = 1
|
|
$$
|
|
|
|
which makes sense, since $p_0$ should have degree zero.
|
|
Therefore, the sequence of polynomials $p_n(\lambda)$ is:
|
|
|
|
$$
|
|
\begin{gather*}
|
|
p_0(\lambda) &=&& && 1
|
|
\\
|
|
p_1(\lambda) &=&& && \lambda
|
|
\\
|
|
p_2(\lambda) &=&& \lambda \lambda - 1
|
|
&=& \lambda^2 - 1
|
|
\\
|
|
p_3(\lambda) &=&& \lambda (\lambda^2 - 1) - \lambda
|
|
&=& \lambda(\lambda^2 - 2)
|
|
\\
|
|
p_4(\lambda) &=&& \lambda (\lambda(\lambda^2 - 2)) - (\lambda^2 - 1)
|
|
&=& \lambda^4 - 3\lambda^2 + 1
|
|
\\
|
|
\vdots & && \vdots && \vdots
|
|
\end{gather*}
|
|
$$
|
|
|
|
But wait, we've seen these before (if you read the previous post, that is).
|
|
These are just the Chebyshev polynomials of the second kind, evaluated at $\lambda / 2$.
|
|
Indeed, their recurrence relations are identical, so the characteristic polynomial of $P_n$ is $U_n(\lambda / 2)$.
|
|
Effectively, this connects an *n*-path to a regular *n+1*-gon.
|
|
|
|
|
|
### Back to Cycles
|
|
|
|
Since the generating function of $U_n$ is known, the generating function for the $c_n$
|
|
(which prompted this) is also easily determined.
|
|
For ease of use, let
|
|
|
|
$$
|
|
P(x; \lambda) = {B(x; \lambda / 2) \over x} = {1 \over 1 - \lambda x +\ x^2}
|
|
$$
|
|
|
|
Discarding the initial $c_0$ and $c_1$ by setting them to zero[^2], the generating function is
|
|
|
|
[^2]: It's a good idea to ask why we can do this.
|
|
Try examining $c_2$ and $c_3$.
|
|
|
|
$$
|
|
\begin{align*}
|
|
c_{n+2}(\lambda) &= \lambda p_{n+1}(\lambda) - 2(p_n(\lambda) + 1)
|
|
\\[14pt]
|
|
{C(x; \lambda) - c_0(\lambda) - x c_1(\lambda) \over x^2}
|
|
&= \lambda \left( {P(x; \lambda) - 1 \over x} \right)
|
|
- 2\left( P(x; \lambda) + {1 \over 1 - x} \right)
|
|
\\
|
|
C &= x \lambda (P - 1) -\
|
|
2x^2\left( P + {1 \over 1 - x} \right)
|
|
\\
|
|
C{(1 - x) \over P} &= x \lambda \left(1 - {1 \over P} \right)(1 - x) -\
|
|
2x^2\left( (1 - x) + {1 \over P} \right)
|
|
\\
|
|
&= x^4 \lambda - 2 x^4 - x^3 \lambda^2 + x^3 \lambda
|
|
+ 2 x^3 + x^2 \lambda^2 - 4 x^2
|
|
\\
|
|
&= x^2 (\lambda - 2) (x^2 - \lambda x - x + \lambda + 2)
|
|
\\[14pt]
|
|
C(x; \lambda) &= x^2 (\lambda - 2)
|
|
{(x^2 - (\lambda + 1) x + \lambda + 2)
|
|
\over (1 - x)(1 - \lambda x + x^2)}
|
|
\end{align*}
|
|
$$
|
|
|
|
While the numerator is considerably more complicated than the one for P,
|
|
the factor $\lambda - 2$ drops out of the entire series.
|
|
This pleasantly informs that 2 is an eigenvalue of all $C_n$.
|
|
|
|
|
|
Off the Beaten Path
|
|
-------------------
|
|
|
|
When we use Laplace expansion on the adjacency matrices, we were very fortunate that the minors
|
|
*also* looked like adjacency matrices undergoing expansion.
|
|
This let us terminate early and recurse.
|
|
From the perspective of the graph, Laplace expansion almost looks like removing a node,
|
|
but requires special treatment for the nodes connected to the one being removed.
|
|
For example, in cycle graphs, the first stage of expansion had three minors:
|
|
|
|
- The node itself, on the main diagonal
|
|
- Being on the main diagonal, this immediately produced another adjacency matrix.
|
|
- Either neighbor connected to it, which are on opposite sides of
|
|
a path after the node is removed
|
|
- Both of these nodes required second expansion to get the *λ*s back on the main diagonal.
|
|
|
|
For "good enough" graphs that are nearly paths (including paths themselves),
|
|
this gives a second-order recurrence relation.
|
|
|
|
|
|
### Trees
|
|
|
|
Another simple family of graphs are [*trees*](https://en.wikipedia.org/wiki/Tree_%28graph_theory%29).
|
|
In some sense, they are the opposite of cycle graphs, since by definition they contain no cycles.
|
|
|
|
Paths are degenerate trees, but we can make them slightly more interesting by instead adding
|
|
exactly one node and edge to (the middle of) a path.
|
|
|
|
{.wide}
|
|
|
|
In this notation, the subscripts denote the consituent paths if the "added" node and the
|
|
one it is connected to are both removed.
|
|
It's easy to see that $T_{a,b} \cong T_{a,b}$, since this just swaps the arms.
|
|
Also, $T_{a, 0} \cong T_{0, a} \cong P_{a + 2}$.
|
|
|
|
Let's try dissecting some of the larger trees.
|
|
The adjacency matrices for $T_{1,3}$ and $T_{2,2}$ are:
|
|
|
|
$$
|
|
\begin{align*}
|
|
T_{1,3} := \begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 3), \\
|
|
(3, 4), \\
|
|
(1, 5)
|
|
]\end{matrix} &\cong
|
|
\begin{pmatrix}
|
|
0 & 1 & 0 & 0 & 0 & 0 \\
|
|
1 & 0 & 1 & 0 & 0 & 1 \\
|
|
0 & 1 & 0 & 1 & 0 & 0 \\
|
|
0 & 0 & 1 & 0 & 1 & 0 \\
|
|
0 & 0 & 0 & 1 & 0 & 0 \\
|
|
0 & 1 & 0 & 0 & 0 & 0
|
|
\end{pmatrix}
|
|
\\ \\
|
|
T_{2,2} := \begin{matrix}[
|
|
(0, 1), \\
|
|
(1, 2), \\
|
|
(2, 3), \\
|
|
(3, 4), \\
|
|
(2, 5)
|
|
]\end{matrix} &\cong
|
|
\begin{pmatrix}
|
|
0 & 1 & 0 & 0 & 0 & 0 \\
|
|
1 & 0 & 1 & 0 & 0 & 0 \\
|
|
0 & 1 & 0 & 1 & 0 & 1 \\
|
|
0 & 0 & 1 & 0 & 1 & 0 \\
|
|
0 & 0 & 0 & 1 & 0 & 0 \\
|
|
0 & 0 & 1 & 0 & 0 & 0
|
|
\end{pmatrix}
|
|
\end{align*}
|
|
$$
|
|
|
|
Starting with $T_{1,3}$, its characteristic polynomial is:
|
|
|
|
$$
|
|
\begin{align*}
|
|
|I \lambda - T_{1,3}|
|
|
&= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 & 0 & 0 \\
|
|
-1 & \lambda & -1 & 0 & 0 & -1 \\
|
|
0 & -1 & \lambda & -1 & 0 & 0 \\
|
|
0 & 0 & -1 & \lambda & -1 & 0 \\
|
|
0 & 0 & 0 & -1 & \lambda & 0 \\
|
|
0 & -1 & 0 & 0 & 0 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&= \lambda (-1)^{6 + 6} m_{6,6} + (-1) (-1)^{2 + 6} m_{2,6}
|
|
\end{align*}
|
|
$$
|
|
|
|
It's easy to see that $m_{6,6}$ is just $p_5(\lambda)$, since the rest of the graph
|
|
other than the additional node is a 5-path.
|
|
But the other minor is trickier.
|
|
|
|
$$
|
|
\begin{align*}
|
|
m_{2,6}
|
|
&= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 & 0 \\
|
|
0 & -1 & \lambda & -1 & 0 \\
|
|
0 & 0 & -1 & \lambda & -1 \\
|
|
0 & 0 & 0 & -1 & \lambda \\
|
|
0 & -1 & 0 & 0 & 0
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&= (-1) (-1)^{5 + 2}
|
|
\left |
|
|
\begin{matrix}
|
|
\lambda & 0 & 0 & 0 \\
|
|
0 & \lambda & -1 & 0 \\
|
|
0 & -1 & \lambda & -1 \\
|
|
0 & 0 & -1 & \lambda \\
|
|
\end{matrix}
|
|
\right |
|
|
\end{align*}
|
|
$$
|
|
|
|
Through one extra expansion, the determinant of this final matrix can be written as
|
|
a product of $\lambda$ and $p_3(\lambda)$.
|
|
|
|
Before making any conjectures, let's do the same thing to $T_{2,2}$.
|
|
|
|
$$
|
|
\begin{align*}
|
|
|I \lambda - T_{2,2}|
|
|
&= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 & 0 & 0 \\
|
|
-1 & \lambda & -1 & 0 & 0 & 0 \\
|
|
0 & -1 & \lambda & -1 & 0 & -1 \\
|
|
0 & 0 & -1 & \lambda & -1 & 0 \\
|
|
0 & 0 & 0 & -1 & \lambda & 0 \\
|
|
0 & 0 & -1 & 0 & 0 & \lambda
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&= (-1)^{6 + 6} \lambda m_{6,6} + (-1) (-1)^{3 + 6} m_{3,6}
|
|
\\ \\
|
|
m_{3,6}
|
|
&= \left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 & 0 \\
|
|
-1 & \lambda & -1 & 0 & 0 \\
|
|
0 & 0 & -1 & \lambda & -1 \\
|
|
0 & 0 & 0 & -1 & \lambda \\
|
|
0 & 0 & -1 & 0 & 0
|
|
\end{matrix}
|
|
\right |
|
|
\\
|
|
&= (-1) (-1)^{5 + 3}
|
|
\left |
|
|
\begin{matrix}
|
|
\lambda & -1 & 0 & 0 \\
|
|
-1 & \lambda & 0 & 0 \\
|
|
0 & 0 & \lambda & -1 \\
|
|
0 & 0 & -1 & \lambda \\
|
|
\end{matrix}
|
|
\right |
|
|
\end{align*}
|
|
$$
|
|
|
|
Here we get something similar: a combination of $p_5(\lambda)$ and an extra term.
|
|
In this case, the final determinant can be written as $p_2(\lambda)^2$.
|
|
|
|
Now it can be observed that the extra terms are the polynomials corresponding
|
|
to $P_a$ and $P_b$ (recall that $p_1(\lambda) = \lambda$, after all).
|
|
In both cases, the second expansion was necessary to get rid of the symmetric -1
|
|
entries added to the matrix.
|
|
The sign of this extra term is always negative, since the -1 entries cancel
|
|
and one of the signs of the minors along the two expansions must be negative.
|
|
|
|
Therefore, the expression for these characteristic polynomials should be:
|
|
|
|
$$
|
|
t_{a,b}(\lambda) = \lambda p_{a + b + 1}(\lambda) - p_a(z) p_b(\lambda)
|
|
$$
|
|
|
|
Note that if *b* is 0, this agrees with the recurrence for $p_n(\lambda)$.
|
|
|
|
|
|
### Examining Small Trees
|
|
|
|
Due to the subscript of the first term of the RHS, this recurrence is harder to turn into
|
|
a generating function.
|
|
Instead, let's look at a few smaller trees to see what kind of polynomials they build.
|
|
We'll also change the variable of the polynomial to *z* for simplicity.
|
|
|
|
The first tree of note is $T_{1,1}$.
|
|
This has characteristic polynomial
|
|
|
|
$$
|
|
\begin{align*}
|
|
t_{1,1}(z) &= z p_{3}(z) - p_1(z) p_1(z)
|
|
\\
|
|
&= z (z^3 - z^2) - z^2
|
|
\\
|
|
&= z^2 (z^2 - 3)
|
|
\end{align*}
|
|
$$
|
|
|
|
Next, we have both $T_{2,1}$ and $T_{1,2}$.
|
|
By symmetry, these are the same graph, so we have characteristic polynomial
|
|
|
|
$$
|
|
\begin{align*}
|
|
t_{1,2}(z) &= z p_{4}(z) - p_1(z) p_2(z)
|
|
\\
|
|
&= z (z^4 - 3z^2 + 2) - z \cdot (z^2 - 1)
|
|
\\
|
|
&= z (z^4 - 4z^2 + 2)
|
|
\end{align*}
|
|
$$
|
|
|
|
Finally, let's look at $T_{1,3}$ and $T_{2,2}$, the trees we used to derive the rule.
|
|
|
|
$$
|
|
\begin{align*}
|
|
t_{1,3}(z) &= z p_{5}(z) - p_1(z) p_3(z)
|
|
\\
|
|
&= z (z^5 - 4z^3 + 3z) - z \cdot (z^3 - 2z)
|
|
\\
|
|
&= z^2 (z^4 - 5z^2 + 5)
|
|
\\[10pt]
|
|
t_{2,2}(z) &= z p_{5}(z) - p_2(z) p_2(z)
|
|
\\
|
|
&= z (z^5 - 4z^3 + 3z) - ( z^2 - 1 )^2
|
|
\\
|
|
&= (z^2 - 1)(z^4 - 4z^2 + 1)
|
|
\end{align*}
|
|
$$
|
|
|
|
Many of these expressions factor surprisingly nicely.
|
|
Further, some of these might seem familiar.
|
|
From the last post, we saw that $z^4 - 5z^2 + 5$ is a factor of $p_9(z)$, from which we know
|
|
it is the minimal polynomial of $2 \cos(\pi / 10)$.
|
|
|
|
This is also true for:
|
|
|
|
- In $t_{1,2}$, the factor $z^4 - 4z^2 + 2$, $p_7(z)$, and $2 \cos(\pi / 8)$, respectively
|
|
- In $t_{2,2}$, the factor $z^4 - 4z^2 + 1$, $p_11(z)$, and $2 \cos(\pi / 12)$, respectively
|
|
|
|
We established that the subscripts of the tree (*a* and *b*) indicate constituent *n*-paths,
|
|
which we know to correspond to *n+1*-gons.
|
|
But these trees also seem to "know" about higher polygons.
|
|
|
|
|
|
### Some Extra Trees
|
|
|
|
$T_{2,3}$ is the first tree not to partition two equal paths or a path and a single node.
|
|
In this regard, the next such tree is $T_{2,4}$.
|
|
|
|
These graphs turn out to have characteristic polynomials whose factors we haven't seen before.
|
|
|
|
$$
|
|
\begin{align*}
|
|
t_{2,3}(z)
|
|
&= z (z^{6} - 6 z^{4} + 9 z^{2} - 3)
|
|
\\
|
|
t_{2,4}(z)
|
|
&= z^{8} - 7 z^{6} + 14 z^{4} - 8 z^{2} + 1
|
|
\end{align*}
|
|
$$
|
|
|
|
Searching the OEIS for the coefficients of $t_{2,4}$ returns sequence
|
|
[A228786](http://oeis.org/A228786), which informs that it is the minimal polynomial
|
|
of $2\sin( \pi/15 )$.
|
|
This sequence also informs that the unknown factor in the other polynomial is
|
|
the minimal polynomial of $2\sin( \pi / 9 )$:
|
|
|
|
In fact, both of these polynomials show up in factorizations of Chebyshev polynomials
|
|
of the *first* kind (specifically, $2T_15(z / 2)$ and $2T_9(z / 2)$).
|
|
Perhaps this is not surprising since we were already working with those of the second kind.
|
|
However, it is interesting to see them appear from the addition of a single node.
|
|
|
|
|
|
Closing
|
|
-------
|
|
|
|
Regardless of whether chains or polygons are more fundamental, it is certainly interesting
|
|
that they are just an algebraic stone's (a *calculus*'s?) toss away from one another.
|
|
Perhaps Euler skipped such stones from the bridges of Koenigsberg which inspired him
|
|
to initiate graph theory.
|
|
|
|
Trees are certainly more complicated than either, and we only investigated those removed
|
|
from a path by a single node.
|
|
Regardless, they still related to Chebyshev polynomials, albeit through their factors.
|
|
|
|
In fact, I was initially prompted to look into them due to a remarkable correspondence between
|
|
certain trees and Platonic solids.
|
|
I have since reorganized these thoughts, as from the perspective of this article, the relationship
|
|
is tangential at best.
|