cube/zenzicubi.co
cube/zenzicubi.co
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

revisions and images to chebyshev.2

Browse Source
This commit is contained in:
queue-miscreant 2025-06-24 06:23:26 -05:00
parent 466a41668b
commit a202facfcd
4 changed files with 670 additions and 197 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
posts/chebyshev/1/index.qmd
View File

@ -1,7 +1,7 @@
---
title: "Generating Polynomials, Part 1: Regular Constructibility"
description: |
"What kinds of regular polygons are constructible with compass and straightedge?"
format:
html:
html-math-method: katex

865
posts/chebyshev/2/index.qmd
View File

@ -1,28 +1,56 @@
---
title: "Generating Polynomials, Part 2: Ghostly Chains"
description: |
"Do polygons without distance still know about planar geometry?"
format:
html:
html-math-method: katex
date: "2021-08-19"
date-modified: "2025-06-20"
categories:
- geometry
- algebra
- graph theory
---
Generating Polynomials, Part 2: Ghostly Chains
==============================================
In the [previous post](), 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?
Fundamentally, it is just a collection of vertices and edges. Each vertex is connected to its two neighbors by two edges. Only examining these figures by their connectedness is precisely the kind of thing *graph theory* deals with. Graph can seem like a strange name, as it has no relation to the familiar graphs on a Cartesian plane one may be familiar with. Either way, it is a worthy subject of study, and will be the focus of this article.
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
----------------------
For polygons in a Euclidean setting, the position of points matters, as well as which points connect to which. A rectangle is different from a trapezoid or a kite. However, topologically, all of these are quadrilateral graphs and cannot be distinguished; they are but a simple notion of 4 points in a loop.
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.
![]()
![
Topologically, all of these are indistinguishable since they all correspond to
the description "4 points in a loop".
](./quadrilaterals.png)
From a graph theory perspective, the vertices are sometimes called nodes. In these graphs, each edge has no direction associated to it, so the graph is called *undirected*. Additionally, these graphs are *planar* since the nodes can be arranged so that no nodes cross another, despite the appearance of the lower-right figure.
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, there's some additional terminology:
If the graph is a simple loop, it is called a [*cycle graph*](https://en.wikipedia.org/wiki/Cycle_graph), denoted $C_n$, where n is the number of nodes. In a cycle graph, all nodes and all edges are identical to each of the others. Therefore, the best geometric interpretation is a shape which is
- Vertices themselves are sometimes instead called *nodes*
- Edge in the graph have no direction associated to them, so the graph is called *undirected*.
- Additionally, these graphs are *planar* since the nodes can be arranged
so that no edges appear to intersect.
- The lower-right figure in the above diagram has intersecting edges,
but the nodes can be rearranged to look like the other graphs, so it is planar.
Graphs themselves typically come in families.
If the graph is a simple loop, 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)
@ -32,105 +60,229 @@ In other words, $C_3$ is analogous to an equilateral triangle, $C_4$ is analogou
### 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.
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 matrix, called an *adjacency matrix*.
Alternatively, these pairs can also be interpreted as addresses in a matrix,
called an *adjacency matrix*.
$$
\begin{align*}
C_3 := \begin{matrix} [(0, 1),\\ (1, 2),\\ (2, 0)] \end{matrix} &\cong
\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \\ \\
C_4 := \begin{matrix} [(0, 1),\\ (1, 2),\\ (2, 3),\\ (3, 0)] \end{matrix} & \cong
\begin{pmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix} \\ \\
C_5 := \begin{matrix} [(0, 1),\\ (1, 2),\\ (2, 3),\\ (3, 4),\\ (4, 0)] \end{matrix} &\cong
\begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{pmatrix} \\
C_3 := \begin{matrix}[
(0, 1), \\
(1, 2), \\
(2, 0)
]\end{matrix} & \cong
\begin{pmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{pmatrix}
\\ \\
C_4 := \begin{matrix} [
(0, 1), \\
(1, 2), \\
(2, 3), \\
(3, 0)
]\end{matrix} & \cong
\begin{pmatrix}
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 0
\end{pmatrix}
\\ \\
C_5 := \begin{matrix}[
(0, 1), \\
(1, 2), \\
(2, 3), \\
(3, 4), \\
(4, 0)
]\end{matrix} &\cong
\begin{pmatrix}
0 & 1 & 0 & 0 & 1 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 0
\end{pmatrix}
\end{align*}
$$
Each adjacency matrix is square, where each column and row refer to a specific node. An entry is 1 when the nodes corresponding to the row and column of its address are joined by an edge (and zero otherwise). For undirected graphs, these matrices are symmetric, since it is possible to traverse an edge in either direction.
Each adjacency matrix is square, where each column and row refer to a specific node.
An entry is 1 when the nodes corresponding to the row and column of its address are joined by an edge (and zero otherwise).
For undirected graphs, these matrices are symmetric, since it is possible to traverse an edge in either direction.
Swapping the labels on two nodes is as simple as exchanging two rows and two columns. Just one of these swaps would flip the sign of the determinant of the adjacency matrix. However, since they occur in pairs, the determinant is invariant of the labelling (equally, a graph invariant).
Swapping the labels on two nodes is as simple as exchanging two rows and two columns.
Just one of these swaps would flip the sign of the determinant of the adjacency matrix.
However, 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), which is among the most mystifying names in math to read for the first time.
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],
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:
[^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:
$$
\begin{gather*}
Ax = \lambda x \implies (\lambda I - A)x = 0 \\ \\
c_5(\lambda) = |\lambda I - C_5| = \left |
\begin{matrix} \lambda & -1 & 0 & 0 & -1 \\ -1 & \lambda & -1 & 0 & 0 \\
0 & -1 & \lambda & -1 & 0 \\ 0 & 0 & -1 & \lambda & -1 \\ -1 & 0 & 0 & -1 & \lambda \end{matrix}
\right | \\
= \lambda m_{1,1}
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 2 \ }}^\text{sign} m_{1, 2}
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 5 \ }}^\text{sign} m_{1, 5}
Ax = \lambda x \implies (\lambda I - A)x = 0
\\ \\
c_5(\lambda) = |\lambda I - C_5|
= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 & -1 \\
-1 & \lambda & -1 & 0 & 0 \\
0 & -1 & \lambda & -1 & 0 \\
0 & 0 & -1 & \lambda & -1 \\
-1 & 0 & 0 & -1 & \lambda
\end{matrix}
\right |
\\
= \lambda m_{1,1}
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 2 \ }}^\text{sign} m_{1, 2}
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 5 \ }}^\text{sign} m_{1, 5}
\end{gather*}
$$
Note that every occurrence of "5" generalizes to higher *n*. The first minor is easily expressed in terms of *another* matrix's characteristic polynomial.
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.
$$
m_{1, 1} = \left |
\begin{matrix} \lambda & -1 & 0 & 0 \\ -1 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\ 0 & 0 & -1 & \lambda \end{matrix}
\right | = |\lambda I - P_4| = p_{5-1}(\lambda)
m_{1, 1}[C_5]
= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 \\
-1 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\
0 & 0 & -1 & \lambda
\end{matrix}
\right |
= |\lambda I - P_4| = p_{5 - 1}(\lambda)
$$
We will come to the meaning of the $P_n$ in a moment. Meanwhile, the second and third minors require another expansion, but one that (thankfully) quickly terminates.
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.
$$
\begin{matrix}
m_{1, 2} =& \left |
\begin{matrix}-1 & -1 & 0 & 0 \\ 0 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\ -1 & 0 & -1 & \lambda \end{matrix}
\right | &=& (-1) \left |
\begin{matrix}\lambda & -1 & 0 \\ -1 & \lambda & -1 \\ 0 & -1 & \lambda \end{matrix}
\right | &+& (-1)(-1)^{1 + 4} \left |
\begin{matrix}-1 & 0 & 0 \\\lambda & -1 & 0 \\ -1 & \lambda & -1 \end{matrix}
\right | \\
&&=& (-1)|\lambda I - P_3| &+& (-1)\overbrace{(-1)^{5}(-1)^{5 - 2}}^{\text{even, even when $\scriptsize n \neq 5$}} \\
&&=& ((-1)p_{5 - 2}(\lambda) &+& (-1)) \\
&&=& -(p_{5 - 2}(\lambda) &+& 1) \\
\\
m_{1, 5} =& \left |
\begin{matrix}-1 & \lambda & -1 & 0 \\ 0 & -1 & \lambda & -1 \\
0 & 0 & -1 & \lambda \\ -1 & 0 & 0 & -1 \end{matrix}
\right | &=& (-1) \left |
\begin{matrix}-1 & \lambda & -1 \\ 0 & -1 & \lambda \\ 0 & 0 & -1 \end{matrix}
\right | &+& (-1)(-1)^{5 - 2} \left |
\begin{matrix}\lambda & -1 & 0 \\ -1 & \lambda & -1 \\
0 & -1 & \lambda \end{matrix}
\right | \\
&&=& (-1)(-1)^{5-2} &+& (-1)(-1)^{5 - 2}|\lambda I - P_3| \\
&&=& (-1)^{5-1}((-1)(-1) &+& (-1)(-1)p_{5 - 2}(\lambda)) \\
&&=& (-1)^{5-1}(1 &+& p_{5 - 2}(\lambda))
m_{1, 2}[C_5]
&= \left |
\begin{matrix}
-1 & -1 & 0 & 0 \\
0 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\
-1 & 0 & -1 & \lambda
\end{matrix}
\right |
&=& (-1)
\left |
\begin{matrix}
\lambda & -1 & 0 \\
-1 & \lambda & -1 \\
0 & -1 & \lambda
\end{matrix}
\right |
&+& (-1)(-1)^{1 + 4}
\left |
\begin{matrix}
-1 & 0 & 0 \\
\lambda & -1 & 0 \\
-1 & \lambda & -1
\end{matrix}
\right |
\\
&&=& (-1)|\lambda I - P_3|
&+& (-1)\overbrace{(-1)^{5}(-1)^{5 - 2}}^{\text{even power, even when $\scriptsize n \neq 5$}}
\\
&&=& (-1)p_{5 - 2}(\lambda) &+& (-1)
\\
&&=& -(p_{5 - 2}(\lambda) &+& 1)
\end{matrix}
$$
All together, this produces a characteristic polynomial in terms of the polynomials $p_n$:
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 $p_n(\lambda)$:
$$
\begin{align*}
&&c_5(\lambda) &= \lambda p_{5 - 1}
+ (-1)(p_{5 - 2} + 1)
+ (-1)\overbrace{(-1)^{5 - 1} (-1)^{5 - 1}}^{\text{even, even when $\scriptsize n \neq 5$}}(p_{5 - 2} + 1) \\
&&&= \lambda p_{5 - 1}
- (p_{5 - 2} + 1)
- (p_{5 - 2} + 1) \\
&&&= \lambda p_{5 - 1}
- 2(p_{5 - 2} + 1) \\
&&\implies c_n(\lambda) &= \lambda p_{n - 1}(\lambda)
- 2(p_{n - 2}(\lambda) + 1) \\
&& c_5(\lambda) &= \lambda p_{5 - 1}
+ (-1)(p_{5 - 2} + 1)
+ (-1)\overbrace{(-1)^{5 - 1} (-1)^{5 - 1}}^{\text{even, even when $\scriptsize n \neq 5$}}(p_{5 - 2} + 1)
\\
&&&= \lambda p_{5 - 1}
- (p_{5 - 2} + 1)
- (p_{5 - 2} + 1)
\\
&&&= \lambda p_{5 - 1}
- 2(p_{5 - 2} + 1)
\\
&& \implies c_n(\lambda) &= \lambda p_{n - 1}(\lambda)
- 2(p_{n - 2}(\lambda) + 1)
\end{align*}
$$
This resembles a recurrence relation, which is great, but it is meaningless without knowing $p_n$.
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
@ -138,190 +290,511 @@ Powerful Chains
The various $P_n$ are in fact the adjacency matrices of a path on *n* nodes.
![]()
<!-- TODO: bad labelling in figure -->
![
Example path graphs of orders 2, 3, and 4
](./path_graphs.png)
$$
\begin{align*}
P_2 &:= \begin{matrix} [(0, 1)] \end{matrix} \cong
\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \\
P_3 &:= \begin{matrix} [(0, 1),\\ (1, 2)] \end{matrix} \cong
\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\
P_4 &:= \begin{matrix} [(0, 1),\\ (1, 2),\\ (2, 3)] \end{matrix} \cong
\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} \\ \\
P_2 &:=
\begin{matrix}[
(0, 1)
]\end{matrix}
\cong \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
\\
P_3 &:=
\begin{matrix}[
(0, 1), \\
(1, 2)
]\end{matrix}
\cong \begin{pmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix}
\\
P_4 &:=
\begin{matrix}[
(0, 1), \\
(1, 2), \\
(2, 3)
]\end{matrix}
\cong \begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}
\end{align*}
$$
Without the entries in the corners of the matrix, the characteristic polynomials of $P_n$ are much easier to solve for.
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.
$$
\begin{gather*}
p_4(\lambda) = |\lambda I - P_4| = \left |
\begin{matrix} \lambda & -1 & 0 & 0 \\ -1 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\ 0 & 0 & -1 & \lambda \end{matrix}
\right | \\ \\
= \lambda \left |
\begin{matrix} \lambda & -1 & 0 \\ -1 & \lambda & -1 \\ 0 & -1 & \lambda \end{matrix}
\right | + (-1)(-1)^{1+2} \left |
\begin{matrix} -1 & -1 & 0 \\ 0 & \lambda & -1 \\ 0 & -1 & \lambda \end{matrix}
\right | \\ \\
= \lambda |\lambda I - P_3| + \left ( (-1) \left |
\begin{matrix} \lambda & -1 \\ -1 & \lambda \end{matrix}
\right | + (-1)(-1) \left |
\begin{matrix} 0 & -1 \\ 0 & \lambda \end{matrix}
\right | \right) \\ \\
= \lambda |\lambda I - P_3| - |\lambda I - P_2| \\
= \lambda p_{4 - 1}(\lambda) - p_{4 - 2}(\lambda) \\
\implies p_{n}(\lambda) = \lambda p_{n - 1}(\lambda) - p_{n - 2}(\lambda)
p_4(\lambda) = |\lambda I - P_4|
= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 \\
-1 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\
0 & 0 & -1 & \lambda
\end{matrix}
\right |
\\ \\
= \lambda \left |
\begin{matrix}
\lambda & -1 & 0 \\
-1 & \lambda & -1 \\
0 & -1 & \lambda
\end{matrix}
\right | + (-1)(-1)^{1+2} \left |
\begin{matrix}
-1 & -1 & 0 \\
0 & \lambda & -1 \\
0 & -1 & \lambda
\end{matrix}
\right |
\\ \\
= \lambda |\lambda I - P_3| + \left (
(-1) \left |
\begin{matrix}
\lambda & -1 \\
-1 & \lambda
\end{matrix}
\right |
+ (-1)(-1) \left |
\begin{matrix}
0 & -1 \\
0 & \lambda
\end{matrix}
\right |
\right)
\\ \\
= \lambda |\lambda I - P_3| - |\lambda I - P_2|
\\
= \lambda p_{4 - 1}(\lambda) - p_{4 - 2}(\lambda)
\\
\implies p_{n}(\lambda) = \lambda p_{n - 1}(\lambda) - p_{n - 2}(\lambda)
\end{gather*}
$$
While the earlier equation for $c_n$ in terms of $p_n$ reminded of a recurrence relation, actually is one (if perhaps eerily familiar).
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
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
$$
|\lambda I - P_2| = \begin{pmatrix}\lambda & -1 \\ -1 & \lambda \end{pmatrix}
|\lambda I - P_2|
= \begin{pmatrix}
\lambda & -1 \\
-1 & \lambda
\end{pmatrix}
= \lambda^2 - 1 = \lambda p_1 - p_0 \\
\implies p_0 = 1
$$
which makes sense, since $p_0$ should have degree zero. Therefore, the sequence of polynomials $p_n(\lambda)$ is:
which makes sense, since $p_0$ should have degree zero.
Therefore, the sequence of polynomials $p_n(\lambda)$ is:
$$
\begin{gather*}
p_0(\lambda) =& && 1 \\
p_1(\lambda) =& && \lambda \\
p_2(\lambda) =& \lambda \lambda - 1 &=& \lambda^2 - 1 \\
p_3(\lambda) =& \lambda (\lambda^2 - 1) - \lambda &=& \lambda(\lambda^2 - 2) \\
p_4(\lambda) =& \lambda (\lambda(\lambda^2 - 2)) - (\lambda^2 - 1)
&=& \lambda^4 - 3\lambda^2 + 1 \\
\vdots&\vdots&&\vdots
p_0(\lambda) &= && 1
\\
p_1(\lambda) &= && \lambda
\\
p_2(\lambda) &= \lambda \lambda - 1
&=& \lambda^2 - 1
\\
p_3(\lambda) &= \lambda (\lambda^2 - 1) - \lambda
&=& \lambda(\lambda^2 - 2)
\\
p_4(\lambda) &= \lambda (\lambda(\lambda^2 - 2)) - (\lambda^2 - 1)
&=& \lambda^4 - 3\lambda^2 + 1
\\
\vdots & \vdots && \vdots
\end{gather*}
$$
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, and the characteristic polynomial of $P_n$ is $U_n(\lambda / 2)$. Effectively, this connects an n-path to a regular $n+1$-gon.
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.
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
### 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 0, the generating function is
Discarding the initial $c_0$ and $c_1$ by setting them to 0[^2], the generating function is
[^2]: It's a good idea to ask why we can do this.
Try examining $c_2$ and $c_3$.
$$
\begin{align*}
c_{n+2}(\lambda) &= \lambda p_{n+1}(\lambda) - 2(p_n(\lambda) + 1) \\ \\
{C(x; \lambda) - c_0(\lambda) - x c_1(\lambda) \over x^2}
&= \lambda \left( {P(x; \lambda) - 1 \over x} \right)
- 2\left( P(x; \lambda) + {1 \over 1 - x} \right) \\
C &= x \lambda (P - 1) -\
2x^2\left( P + {1 \over 1 - x} \right) \\
C{(1 - x) \over P} &= x \lambda \left(1 - {1 \over P} \right)(1 - x) -\
2x^2\left( (1 - x) + {1 \over P} \right) \\
&= x^4 \lambda - 2 x^4 - x^3 \lambda^2 + x^3 \lambda
+ 2 x^3 + x^2 \lambda^2 - 4 x^2 \\
&= x^2 (\lambda - 2) (x^2 - \lambda x - x + \lambda + 2) \\ \\
C(x; \lambda) &= x^2 (\lambda - 2)
{(x^2 - (\lambda + 1) x + \lambda + 2)
\over (1 - x)(1 - \lambda x + x^2)}
c_{n+2}(\lambda) &= \lambda p_{n+1}(\lambda) - 2(p_n(\lambda) + 1)
\\ \\
{C(x; \lambda) - c_0(\lambda) - x c_1(\lambda) \over x^2}
&= \lambda \left( {P(x; \lambda) - 1 \over x} \right)
- 2\left( P(x; \lambda) + {1 \over 1 - x} \right)
\\
C &= x \lambda (P - 1) -\
2x^2\left( P + {1 \over 1 - x} \right)
\\
C{(1 - x) \over P} &= x \lambda \left(1 - {1 \over P} \right)(1 - x) -\
2x^2\left( (1 - x) + {1 \over P} \right)
\\
&= x^4 \lambda - 2 x^4 - x^3 \lambda^2 + x^3 \lambda
+ 2 x^3 + x^2 \lambda^2 - 4 x^2
\\
&= x^2 (\lambda - 2) (x^2 - \lambda x - x + \lambda + 2)
\\ \\
C(x; \lambda) &= x^2 (\lambda - 2)
{(x^2 - (\lambda + 1) x + \lambda + 2)
\over (1 - x)(1 - \lambda x + x^2)}
\end{align*}
$$
While the numerator is considerably more complicated than the one for P, the factor $\lambda -\ 2$ drops out of the entire series, pleasantly informing that 2 is an eigenvalue of all $C_n$.
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 $C_n$.
Some other Exceptional Graphs
-----------------------------
Off the Beaten Path
-------------------
And now for something completely different (and which prompted me to research this topic in the first place). Another simple family of graphs are [*trees*](https://en.wikipedia.org/wiki/Tree_%28graph_theory%29). They are in some sense opposite to the cycle graphs, since they contain no cycles.
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.
### Platonic Trees
### Trees
Of trees, three small ones (which are not also paths) stand out. They are based on the [Schläfli symbols](https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol) of the Platonic solids. These expressions convey which regular polygon is present, and how many of them are found at every vertex.
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.
- Tetrahedron: {3, 3}
- Equilateral triangles, three around a vertex
- Octahedron: {3, 4}
- Equilateral triangles, four around a vertex
- Icosahedron: {3, 5}
- Equilateral triangles, five around a vertex
- Cube: {4, 3}
- Squares, three around a vertex
- Dodecahdedron: {5, 3}
- Regular pentagons, three around a vertex
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.
Each edge in a Platonic solid connects two faces, which gives a third parameter of 2. From each parameter, create a $n-1$-path, and join the three to a central node. Intuitively, this is because we have shown that $n-1$-paths are related to regular *n*-gons, which show up in the pairs of dual Platonic solids.
<!-- TODO: new image for T_{1,1} through T_{2,2} -->
$$
\begin{align*}
T_{1,1} := \begin{matrix}[
(0, 1), \\
(1, 2), \\
(1, 3)
]\end{matrix} & \cong
\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0
\end{pmatrix}
\\ \\
T_{1,2} := \begin{matrix} [
(0, 1), \\
(1, 2), \\
(2, 3), \\
(1, 4)
]\end{matrix} & \cong
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0
\end{pmatrix}
\\ \\
T_{1,3} := \begin{matrix}[
(0, 1), \\
(1, 2), \\
(2, 3), \\
(3, 4), \\
(1, 5)
]\end{matrix} &\cong
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0
\end{pmatrix}
\\ \\
T_{2,2} := \begin{matrix}[
(0, 1), \\
(1, 2), \\
(2, 3), \\
(3, 4), \\
(2, 5)
]\end{matrix} &\cong
\begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix}
\end{align*}
$$
::: {}
![]()
Adjacency matrices of T as a heatmap. Purple is 0, yellow is 1.
:::
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 one of the larger trees.
Dispensing with the Laplace expansion and finding a recurrence relation, the characteristic polynomials for each of the three trees is
$$
\begin{align*}
|\lambda I - T_{3,3}| = t_{3,3}(\lambda)
&= \lambda^{6} - 5 \lambda^{4} + 5 \lambda^{2} - 1 \\
&= (\lambda - 1) (\lambda + 1) (\lambda^{4} - 4 \lambda^{2} + 1) \\
|\lambda I - T_{3,4}| = t_{3,4}(\lambda)
&= \lambda^{7} - 6 \lambda^{5} + 9 \lambda^{3} - 3 \lambda \\
&= \lambda (\lambda^{6} - 6 \lambda^{4} + 9 \lambda^{2} - 3) \\
|\lambda I - T_{3,5}| = t_{3,5}(\lambda)
&= \lambda^{8} - 7 \lambda^{6} + 14 \lambda^{4} - 8 \lambda^{2} + 1
|I \lambda - T_{1,3}|
&= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 & 0 & 0 \\
-1 & \lambda & -1 & 0 & 0 & -1 \\
0 & -1 & \lambda & -1 & 0 & 0 \\
0 & 0 & -1 & \lambda & -1 & 0 \\
0 & 0 & 0 & -1 & \lambda & 0 \\
0 & -1 & 0 & 0 & 0 & \lambda
\end{matrix}
\right |
\\
&= (-1)^{6 + 6} \lambda m_{6,6} + (-1)^{2 + 6} (-1) m_{2,6}
\end{align*}
$$
Searching the OEIS for the coefficients of $t_{3,5}$ 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, where $m_n$ is the minimal polynomial of $2\sin( \pi / n )$:
- $t_{3,3} = m_6(\lambda) \cdot -m_6(-\lambda) \cdot m_{12}(\lambda)$
- $t_{3,4} = m_1(\lambda) \cdot m_9(\lambda)$
Perhaps this is not surprising, given how $2\cos(\pi / n)$ also appear in the spectra of graphs. However, I am perplexed by the apparent relationship of tetrahedra to dodecagons, octahedra to enneagons, and icosahedra to 15-gons. It is also strange that 9 is *not* related to the smallest Platonic solid, and is sandwiched between 12 and 15.
### Wooden Tiles
Platonic solids can be seen as regular tilings of the sphere. Schläfli symbols also exist for the regular tilings of the plane (which can also tile the torus, another topologically significant object).
- Triangular tiling: {3, 6}
- Equilateral triangles; 6 around a vertex
- Square tiling: {4, 4}
- Squares; 4 around a vertex
- Hexagonal tiling: {6, 3}
- Regular hexagons; 3 around a vertex
::: {}
![]()
Adjacency matrices as a heatmap
:::
I would prefer *not* to wrestle the adjacency matrices into a general recurrence relation, but if one would like to go down this route, try using the Laplace expansion of the southmost tip of the "island" in the $T_{3,n}$ graphs as an example.
The characteristic polynomials of *these* graphs are
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.
$$
\begin{align*}
|\lambda I - T_{4,4}| = t_{4,4}(\lambda)
&= \lambda^{8} - 7 \lambda^{6} + 14 \lambda^{4} - 8 \lambda^{2} \\
&= (\lambda - 1) (\lambda + 1) (\lambda^{4} - 4 \lambda^{2} + 1) \\
&= m_6(\lambda) \cdot -m_6(-\lambda) \cdot m_8(\lambda) \\
|\lambda I - T_{3,6}| = t_{3,6}(\lambda)
&= \lambda^{9} - 8 \lambda^{7} + 20 \lambda^{5} - 17 \lambda^{3} + 4 \lambda \\
&= \lambda (\lambda - 1)(\lambda + 1) (\lambda - 2)(\lambda + 2)
(\lambda^{2} - \lambda - 1) (\lambda^{2} + \lambda - 1) \\
&= m_1(\lambda) \cdot m_6(\lambda) \cdot -m_6(-\lambda)
\cdot m_2(\lambda) \cdot -m_2(-\lambda)
\cdot m_{10}(\lambda) \cdot m_{10}(-\lambda) \\
m_{2,6}
&= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 & 0 \\
0 & -1 & \lambda & -1 & 0 \\
0 & 0 & -1 & \lambda & -1 \\
0 & 0 & 0 & -1 & \lambda \\
0 & -1 & 0 & 0 & 0
\end{matrix}
\right |
\\
&= (-1)^{5 + 2} (-1)
\left |
\begin{matrix}
\lambda & 0 & 0 & 0 \\
0 & \lambda & -1 & 0 \\
0 & -1 & \lambda & -1 \\
0 & 0 & -1 & \lambda \\
\end{matrix}
\right |
\end{align*}
$$
Where the $m_n$ come from the sine polynomial series. To me, this implies that where chains are connected to cosines of fractions of a half-turn, these trefoil trees are connected to sines of fractions of turns (but you'd have to derive a general rule to be sure).
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 $T_{2,2}$.
$$
\begin{align*}
|I \lambda - T_{2,2}|
&= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 & 0 & 0 \\
-1 & \lambda & -1 & 0 & 0 & 0 \\
0 & -1 & \lambda & -1 & 0 & -1 \\
0 & 0 & -1 & \lambda & -1 & 0 \\
0 & 0 & 0 & -1 & \lambda & 0 \\
0 & 0 & -1 & 0 & 0 & \lambda
\end{matrix}
\right |
\\
&= (-1)^{6 + 6} \lambda m_{6,6} + (-1)^{3 + 6} (-1) m_{3,6}
\\ \\
m_{3,6}
&= \left |
\begin{matrix}
\lambda & -1 & 0 & 0 & 0 \\
-1 & \lambda & -1 & 0 & 0 \\
0 & 0 & -1 & \lambda & -1 \\
0 & 0 & 0 & -1 & \lambda \\
0 & 0 & -1 & 0 & 0
\end{matrix}
\right |
\\
&= (-1)^{5 + 3} (-1)
\left |
\begin{matrix}
\lambda & -1 & 0 & 0 \\
-1 & \lambda & 0 & 0 \\
0 & 0 & \lambda & -1 \\
0 & 0 & -1 & \lambda \\
\end{matrix}
\right |
\end{align*}
$$
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 $P_b$ ($p_1(\lambda) = \lambda$, 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:
$$
t_{a,b}(z) = z \cdot p_{a + b + 1}(z) - p_a(z) p_b(z)
$$
Note that if *b* is 0, this coincides with the recurrence for $p_n(z)$.
### 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.
The first tree of note is $T_{1,1}$.
This has characteristic polynomial
$$
\begin{align*}
t_{1,1}(z) &= z \cdot p_{3} - p_1(z) p_1(z)
\\
&= z (z^3 - z^2) - z^2
\\
&= z^2 (z^2 - 3)
\end{align*}
$$
Next, we have both $T_{2,1}$ and $T_{1,2}$.
By symmetry, these are the same graph, so we have characteristic polynomial
$$
\begin{align*}
t_{1,2}(z) &= z \cdot p_{4} - p_1(z) p_2(z)
\\
&= z (z^4 - 3z^2 + 2) - z (z^2 - 1)
\\
&= z (z^4 - 4z^2 + 2)
\end{align*}
$$
Finally, let's look at $T_{1,3}$ and $T_{2,2}$, the trees we used to derive the rule.
$$
\begin{align*}
t_{1,3}(z) &= z \cdot p_{5} - p_1(z) p_3(z)
\\
&= z (z^5 - 4z^3 + 3z) - z \cdot (z^3 - 2z)
\\
&= z^2 (z^4 - 5z^2 + 5)
\\[10pt]
t_{2,2}(z) &= z \cdot p_{5} - p_2(z) p_2(z)
\\
&= z (z^5 - 4z^3 + 3z) - ( z^2 - 1 )^2
\\
&= (z^2 - 1)(z^4 - 4z^2 + 1)
\end{align*}
$$
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 $t_{1,2}$,
- $z^4 - 4z^2 + 2$, $p_7(z)$, and $2 \cos(\pi / 8)$, respectively
- In $t_{2,2}$,
- $z^4 - 4z^2 + 1$, $p_11(z)$, 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 other factor is the minimal polynomial of $2\sin( \pi / 9 )$:
In fact, both of these polynomials show up in the factorization
of Chebyshev polynomials of the *first* kind.
Perhaps this is not surprising, given how $2\cos(\pi / n)$ also appear in the spectra of graphs.
However, it is immensely interesting to see them pop out 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.
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 reorganized these thoughts, since from the perspective of this article, the relationship
is tangential at best.

BIN
posts/chebyshev/2/path_graphs.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 23 KiB

BIN
posts/chebyshev/2/quadrilaterals.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 83 KiB