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edited Jul 1, 2022 at 15:45

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This was too long for a comment.

The chemistry background is basically this: an "aromatic system" (a certain type of molecule) with 2n carbon atoms has 2n "molecular orbitals" where electrons can go, each of which holds two electrons. These orbitals are linear combinations of one "atomic orbital" from each of the carbon atoms, which happen to line up in the right way that they combine, and are spread out over the entire molecule . (Yes, this is a chemist's approximation to certain things in quantum mechanics.)

Their energies are given by the eigenvalues of the adjacency matrix (interpreted in the right units). There are 2n electrons that will go in them. Two electrons (with opposite spins) can go in each orbital, and they fill the orbitals from lowest energy to highest. So the $n$th eigenvalue is the energy of the highest occupied molecular orbital (HOMO), and the $n+1$st is the energy of the lowest unoccupied molecular orbital (LUMO).

In particular, it is probably reasonable to assume for chemical purposes that all the vertices of the graph have degree 2 or 3; think of systems like the ones illustrated in this Wikipedia article the ones illustrated in this Wikipedia article.

This was too long for a comment.

The chemistry background is basically this: an "aromatic system" (a certain type of molecule) with 2n carbon atoms has 2n "molecular orbitals" where electrons can go, each of which holds two electrons. These orbitals are linear combinations of one "atomic orbital" from each of the carbon atoms, which happen to line up in the right way that they combine, and are spread out over the entire molecule . (Yes, this is a chemist's approximation to certain things in quantum mechanics.)

Their energies are given by the eigenvalues of the adjacency matrix (interpreted in the right units). There are 2n electrons that will go in them. Two electrons (with opposite spins) can go in each orbital, and they fill the orbitals from lowest energy to highest. So the $n$th eigenvalue is the energy of the highest occupied molecular orbital (HOMO), and the $n+1$st is the energy of the lowest unoccupied molecular orbital (LUMO).

In particular, it is probably reasonable to assume for chemical purposes that all the vertices of the graph have degree 2 or 3; think of systems like the ones illustrated in this Wikipedia article.

This was too long for a comment.

The chemistry background is basically this: an "aromatic system" (a certain type of molecule) with 2n carbon atoms has 2n "molecular orbitals" where electrons can go, each of which holds two electrons. These orbitals are linear combinations of one "atomic orbital" from each of the carbon atoms, which happen to line up in the right way that they combine, and are spread out over the entire molecule . (Yes, this is a chemist's approximation to certain things in quantum mechanics.)

Their energies are given by the eigenvalues of the adjacency matrix (interpreted in the right units). There are 2n electrons that will go in them. Two electrons (with opposite spins) can go in each orbital, and they fill the orbitals from lowest energy to highest. So the $n$th eigenvalue is the energy of the highest occupied molecular orbital (HOMO), and the $n+1$st is the energy of the lowest unoccupied molecular orbital (LUMO).

In particular, it is probably reasonable to assume for chemical purposes that all the vertices of the graph have degree 2 or 3; think of systems like the ones illustrated in this Wikipedia article.

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created Mar 6, 2010 at 13:06

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This was too long for a comment.

The chemistry background is basically this: an "aromatic system" (a certain type of molecule) with 2n carbon atoms has 2n "molecular orbitals" where electrons can go, each of which holds two electrons. These orbitals are linear combinations of one "atomic orbital" from each of the carbon atoms, which happen to line up in the right way that they combine, and are spread out over the entire molecule . (Yes, this is a chemist's approximation to certain things in quantum mechanics.)

Their energies are given by the eigenvalues of the adjacency matrix (interpreted in the right units). There are 2n electrons that will go in them. Two electrons (with opposite spins) can go in each orbital, and they fill the orbitals from lowest energy to highest. So the $n$th eigenvalue is the energy of the highest occupied molecular orbital (HOMO), and the $n+1$st is the energy of the lowest unoccupied molecular orbital (LUMO).

In particular, it is probably reasonable to assume for chemical purposes that all the vertices of the graph have degree 2 or 3; think of systems like the ones illustrated in this Wikipedia article.