# Classes of Graphs used in Chemistry

I am looking for common categories of graphs used in chemistry, for math research I am doing in graph theory.

When I write categories, the meaning is not for graphs design or types of design, but for categories of nature-originated large graphs (like large molecules), that can be mathematically described. For example, trees, unicyclic graphs, bipartite graphs, etc.

I mention it because I saw a similar question on the Biology Stack Exhange, and the response there is not the meaning of the asker at all.

I tried to look for scientific articles that review it but did not find any related survey.

• Well, typical chemist or biologist doesn't even know what graph theory is :( It's used pretty much only by guys bringing chemistry and maths together... and there's no tag for it ;) Jun 2 '21 at 18:17
• Graphs as an underlying concept to store and retrieve molecular geometries (incl. retrieve of sub-sturctures) in (reaction) databases like here and here? Jun 2 '21 at 19:28
• Outside of molecular geometries you might look at Diagrammatic Perturbation Theory, Feynman diagrams being the best known. physics.stackexchange.com/questions/41848/… has a link that might be of interest Jun 2 '21 at 19:41
• In biochemistry, there are graphs describing metabolic pathways.
– Karsten Theis
Jun 2 '21 at 20:06
• Another use of graph theory is in approximate solutions to the Ornstein-Zernike equation, which occurs in the theory of the structure of fluids Jun 2 '21 at 20:44

### Fullerene graphs

One example of a category of graphs that is used in chemistry is the set of Fullerene graphs. One Fullerene graph that is especially interesting to some chemists, is the 26-Fullerene graph. You may also be interested in this paper about Fullerene graphs, or the book An Atlas of Fullerenes by theoretical chemists Patrick Fowler and David Manolopoulos.

## Hückel method can be related to Graph theory

Graph theory is not part of the average Chemistry undergraduate curriculum, so I have just cursory knowledge about it. But from what I read about, it shows everywhere in Chemistry. Regarding Matter Modeling, Hückel molecular orbital theory is one of the early wins of quantitative theoretical chemistry, developed by Erich Hückel in the 1930s, to explain systems containing conjugated double bonds that show up a lot in organic chemistry. For a bit of history on it, see (1). Benzene is the poster child of these systems.

Hückel proposed the energy levels in these systems can be obtained by solving a secular determinant with this form:

$$det([H_{ij} - ES_{ij}]) = 0$$

Where $$H_{ij}$$ and $$S_{ij}$$ are known as the overlap and Hamiltonian matrices. To solve it, he adopted the following approximations:

$$S_{ij} = 1,\text{ if } i = j$$ $$S_{ij} = 0,\text{ if } i \neq j$$ $$H_{ij} = \alpha,\text{ if } i = j$$ $$H_{ij} = \beta,\text{ if } i \text{ adjacent to } j$$ $$H_{ij} = 0,\text{ otherwise}$$

Where $$\alpha$$ is approximately the energy of a electron in a $$2p_z$$ isolated carbon orbital, and $$\beta$$ approximately the interaction energy of parallel $$2p_z$$ orbitals in adjacent carbons.

Taking the example of benzene, we obtain this:

Then we can use a computer algebra system (CAS) like SymPy to solve this secular determinant:

from sympy import symbols, Matrix, roots

E, α, β = symbols('E α β')
M = Matrix([[α-E,β,0,0,0,β],
[β,α-E,β,0,0,0],
[0,β,α-E,β,0,0],
[0,0,β,α-E,β,0],
[0,0,0,β,α-E,β],
[β,0,0,0,β,α-E]])
benzene_polynomial = M.det()
benzene_roots = roots(benzene_polynomial, E)
print('Solutions:', benzene_roots)


Getting the following result:

Solutions: {α + 2*β: 1, α - 2*β: 1, α + β: 2, α - β: 2}


Filling the pi electrons contributed by each of the six carbons according to the Aufbau principle, we get:

That explains the stability of benzene, as every pi electron ends in a orbital at a energy level bellow the reference energy $$\alpha$$ (note $$\beta$$ is a negative number), and they're all paired up, thus avoiding a reactive radical form with unpaired electrons. So we find benzene and benzene-like molecules everywhere in organic chemistry. The nucleobases in DNA, for example, have benzene-like rings in their structures.

As explained in (2), the Huckel secular determinant can be related to the adjacency matrix of the molecular graph:

Theorem 4. If λ is an eigenvalue and z an eigenvector of the matrix A, then α + βλ is an eigenvalue and z is an eigenvector of the matrix H.

From this theorem it follows that the Hückel molecular orbitals coincide with the eigenvectors zj of eigenvalues λj, j=1,2,...,n, of the adjacency matrix of the Hückel graph. The eigenvalues λj of the matrix A and the energies Ej of the corresponding electrons are related simply as

Ej = α + βλj.

There are exactly n different molecular orbitals, namely the zj for j=1,2,...,n. This important conclusion shows that there is a deep and far-reaching relation between the Hückel molecular orbital theory and graph spectral theory. The Hückel theory provides an important field of application of the graph spectra. The main contribution of [9] was to establish that the Hückel theory and theory of graph spectra are essentially the same.

References: