## Isomers of C6H14

In researching the applications of graph theory, Dr. Sharma asked me to consider the graphs of isomers of Carbon-Hydrogen compounds. We derived the formula for possible molecules. If “c” is the number of Carbon atoms and “h” is the number of Hydrogen atoms, then h=2c+2

So if you are given 6 Carbon atoms, it is possible to form a C6H14 molecule. With these fixed number of atoms, there are several possible structures of the graphs. These graphs are called isomers.

In the graphs of Carbon-Hydrogen compounds, each Carbon is connected to 4 other atoms, but each Hydrogen can only connect to one atom. Therefore, since the graphs of molecules must be connected, Hydrogen atoms only connect to Carbon atoms. So the true underlying structure of Carbon-Hydrogen compounds comes from the construction of the Carbon atoms.

Therefore, drawing all non-isomorphic trees on 6 vertices, where the highest degree of any vertex is 4, creates the underlying structure of all possible C6H14 isomers. Below are pictures of the possible isomer constructions. The true construction of each isomer can be formed by attaching a Hydrogen atom to each Carbon atom until the degree of each Carbon atom is 4.

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