
Applications of Graph Theory (2007) - aoldoni
http://www.dharwadker.org/pirzada/applications/
======
1971genocide
As a mathematical peasant who does modelling.

# If you have a small data set - you set notation.

# If you have 100s of larger data set - try to formulate your problem as a
graph theory problem - Its easy to walk small graphs.

# If you have 10,000 data points - start to think of your problem in terms of
matrix.

I know there is going to strong disagreement about my rule of thumb but I
found it useful for many problems.

~~~
sdenton4
Fair enough. Graph theory has a collection of quite well-known algorithms that
run fast, which mostly come down to clever manipulations of depth-first and
breadth-first search algorithms. These typically run in O(n+k), where n is the
number of vertices and k is the number of edges. In large networks, it's still
often the case that k is proportional to n (most people have less than 1000
facebook friends, so we can bound k by 1000n...), so this is still linear in
n.

I tend to see lots of graph algorithms which run in O(n^3) time, which tends
to be fine for medium-sized data sets, but yeah, breaks down when you get into
the 10,000+ sized data sets.

And then there's 'matrix' methods you mention. These can often be tied to an
area called algebraic graph theory, which looks at properties of matrices
(like the adjacency matrix) to try to pull out information about the graph.
Relaxing the graph problem and allowing the full range of matrix operations
allows some fast linear algebraic methods to be used, though the final
solution may need some massaging to make sense, and may not be perfectly
optimal.

A great example of this is spectral clustering, which uses eigenvectors of a
relative of the adjacency matrix to perform a clustering of the graph's nodes.
The optimization that the clustering is solving is actually an (iirc) NP-hard
problem; casting it as an eigenvalue problem gives an O(n^2) approximation of
a solution, which then gets hammered into something we can live with...

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amelius
When reading the title, I was thinking more of real-world applications. For
instance, the Cantor-Schröder-Bernstein theorem or the "Knight tour's"
problems seem like rather abstract problems.

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johnaspden
Wow! I tried to prove Cantor-Schroder-Bernstein last year and couldn't get my
head round it. This makes it obvious!

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lifeisstillgood
My positive takeaway from this is that "graph theory" which I see as a single
lump of knowledge (that I don't have), is actually still developing and being
used by practitioners - the paper several times in the summary mentions
"rapidly growing fields etc). To me that signals the use of this is changing,
and being a dumb ass who knows nothing is not so bad

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sotojuan
A bit off topic, but what are some good resources for studying graph theory?

~~~
pmiller2
I strongly recommend Introduction to Graph Theory, 2nd edition, by Doug West.
It covers the basics right up through graduate-level material, and has
excellent exercises.

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antonio-rt
[http://barabasi.com/networksciencebook/](http://barabasi.com/networksciencebook/)

------
gphilip
From the home page ([http://www.dharwadker.org/](http://www.dharwadker.org/)):

A new proof of the Four Colour Theorem, by Ashay Dharwadker.

Abstract

#########################################

We present a new proof of the famous four colour theorem using algebraic and
topological methods. Recent research in physics shows that this proof directly
implies the Grand Unification of the Standard Model with Quantum Gravity in
its physical interpretation and conversely the existence of the standard model
of particle physics shows that nature applies this proof of the four colour
theorem at the most fundamental level, giving us a grand unified theory. In
particular, we have shown how to use this theory to predict the Higgs Boson
Mass [arXiv:0912.5189] with precision. Thus, nature itself demonstrates the
logical completeness and consistency of the proof. This proof was first
announced by the Canadian Mathematical Society in 2000. The proof appears as
the twelfth chapter of the text book Graph Theory published by Orient Longman
and Universities Press of India in 2008. This proof has also been published in
the Euroacademy Series Baltic Horizons No. 14 (111) dedicated to Fundamental
Research in Mathematics in 2010. Finally, the proof features in an exquisitely
illustrated edition of The Four Colour Theorem published by Amazon in 2011.
The Endowed Chair of the Institute of Mathematics in recognition of this
achievement was bestowed in 2012.

#########################################

See also
([http://www.dharwadker.org/standard_model/](http://www.dharwadker.org/standard_model/)):

Title: Grand Unification of the Standard Model with Quantum Gravity Author:
Ashay Dharwadker

Abstract:

#########################################

We show that the mathematical proof of the four colour theorem [1] directly
implies the existence of the standard model, together with quantum gravity, in
its physical interpretation. Conversely, the experimentally observable
standard model and quantum gravity show that nature applies the mathematical
proof of the four colour theorem, at the most fundamental level. We preserve
all the established working theories of physics: Quantum Mechanics, Special
and General Relativity, Quantum Electrodynamics (QED), the Electroweak model
and Quantum Chromodynamics (QCD). We build upon these theories, unifying all
of them with Einstein's law of gravity. Quantum gravity is a direct and
unavoidable consequence of the theory. The main construction of the Steiner
system in the proof of the four colour theorem already defines the
gravitational fields of all the particles of the standard model. Our first
goal is to construct all the particles constituting the classic standard
model, in exact agreement with 't Hooft's table [8]. We are able to predict
the exact mass of the Higgs particle and the CP violation and mixing angle of
weak interactions. Our second goal is to construct the gauge groups and
explicitly calculate the gauge coupling constants of the force fields. We show
how the gauge groups are embedded in a sequence along the cosmological
timeline in the grand unification. Finally, we calculate the mass ratios of
the particles of the standard model. Thus, the mathematical proof of the four
colour theorem shows that the grand unification of the standard model with
quantum gravity is complete, and rules out the possibility of finding any
other kinds of particles.

#########################################

~~~
thom
There's a fascinating history of Usenet beef around this:
[http://www.log24.com/log05/050725-Crank.html](http://www.log24.com/log05/050725-Crank.html).

