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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math
Subject: Re: Hairy combinatorics problem.
Message-ID: <365@ihnet.UUCP>
Date: Mon, 17-Feb-86 11:04:41 EST
Article-I.D.: ihnet.365
Posted: Mon Feb 17 11:04:41 1986
Date-Received: Tue, 18-Feb-86 04:01:37 EST
References: <702@harvard.UUCP>
Organization: AT&T Bell Laboratories
Lines: 42

> I wish to count the number of acyclic, saturated hydrocarbons with n carbons.
> These molecules can be thought of as acyclic, undirected, connected graphs,
> in which the carbons are nodes, the carbon-carbon bonds are edges, and the
> hydrogens are ignored.  

Problems like this can be quite frustrating,
because I feel like I should be able to solve them.
We did similar problems in a graph theory course I
took 4 years ago, but I guess project planning and methodology
have withered my brain since then.
It has something to do with manipulating generating functions
for rooted instances of the carbon trees.

At any rate, I am not being mathematical today.
Instead, I am wearing my chemistry hat,
something I rarely do, since I have only taken one
college level chemistry course.  But here goes.
Representing hydrocarbons as trees with degree  <=  4 breaks down for
modest N (say N>15).
Yes, each hydrocarbon has a corresponding carbon tree,
but many trees do not correspond to physical molecules.
To illustrate, begin with a methane molecule,
represented by the degenerate tree (one point).
Now replace each hydrogen atom with a methyl group,
producing a pentane isomer.
The tree is a 4-way star.
Again, replace each hydrogen with a methyl group,
making a 17-carbon glob.
Repeat od-imnosium.
The graphical model gives no hint of trouble,
yet the number of carbon atoms grows exponentially, while the size of the
molecule (maximum radius) increases linearly.
We have a problem, right?!?
Even the 17-carbon glob may not be possible.
Don't have my table of bond angles and distances handy.
In fact, I conjecture, as N approaches infinity,
the probability that a random N-node tree will correspond to a physically
realizable hydrocarbon approaches zero.
Well anyways, it's still a fun problem.
-- 
	Why don't we do it in the road?
			Karl Dahlke    ihnp4!ihnet!eklhad