Re: Quantum Gravity 199.1: Riccati DE and The Chromatic Polynomial and Chromatic Number
- From: "hhc314@xxxxxxxxx" <hhc314@xxxxxxxxx>
- Date: Sun, 04 Nov 2007 12:46:04 -0800
On Nov 3, 1:06 pm, OsherD <mdocto...@xxxxxxxxx> wrote:
From Osher Doctorow
It turns out that the chromatic polynomial g(z) or g_G(z) where G is a
graph, which counts the number of ways to color G with exactly
colors, is usually a multiple of the right hand side of the Riccati
Differential equation and in particular the Logistic Differential
Equation subtype of the Riccati DE. The "kernel" as I call it is the
chromatic polynomial for complete graph K2 in Wolfram's notation:
1) g_K2(z) = (z - 1)z = z^2 - z
which of course is the right hand side of the Logistic Differential
Equation:
2) dy/dt = ky(1 - y)
with k = -1. Kn can be depicted as the regular polygon with n edges,
n = 1, 2, 3, .... , except that every other pair of sequential edges
are connected by a "diagonal" line segment and so are opposite
edges. It is the "kernel" of which g_G(z) is a multiple, that is to
say when g_G(z) is divided by the "kernel" then the result is a
polynomial. For example, for K3 the corresponding g_K(z) is:
3) g_K3(z) = z(z - 1)(z - 2)
This is a good way to learn graphs, by the way, including the Cyclic
graphs Cn, the Star graphs Sn, the Wheel graph Wn, most of which have
g(z) a multiple of the above kernel.
Fendley and Krushkal's paper cited earlier generalizes this
considerably in certain directions. Notice also that it mentions that
the chromatic polynomial of planar graphs relates to the golden ratio,
which in turn we know from earlier threads relates to the Riccati
Differential Equation.
Osher Doctorow
Sadly, just more Osher nonsense. No physics content.
Harry C.
.
- Follow-Ups:
- References:
- Prev by Date: Re: How do the brain neurons compute?
- Next by Date: Re: How do the brain neurons compute?
- Previous by thread: Quantum Gravity 199.1: Riccati DE and The Chromatic Polynomial and Chromatic Number
- Next by thread: Re: Quantum Gravity 199.1: Riccati DE and The Chromatic Polynomial and Chromatic Number
- Index(es):
Relevant Pages
|
