Graph Theoretical Representation of Atomic Asymmetry and Molecular Chirality of Benzenoids in Two-Dimensional Space

In order to explore atomic asymmetry and molecular chirality in 2D space, benzenoids composed of 3 to 11 hexagons in 2D space were enumerated in our laboratory. These benzenoids are regarded as planar connected polyhexes and have no internal holes; that is, their internal regions are filled with hexagons. The produced dataset was composed of 357,968 benzenoids, including more than 14 million atoms. Rather than simply labeling the huge number of atoms as being either symmetric or asymmetric, this investigation aims at exploring a quantitative graph theoretical descriptor of atomic asymmetry. Based on the particular characteristics in the 2D plane, we suggested the weighted atomic sum as the descriptor of atomic asymmetry. This descriptor is measured by circulating around the molecule going in opposite directions. The investigation demonstrates that the weighted atomic sums are superior to the previously reported quantitative descriptor, atomic sums. The investigation of quantitative descriptors also reveals that the most asymmetric atom is in a structure with a spiral ring with the convex shape going in clockwise direction and concave shape going in anticlockwise direction from the atom. Based on weighted atomic sums, a weighted F index is introduced to quantitatively represent molecular chirality in the plane, rather than merely regarding benzenoids as being either chiral or achiral. By validating with enumerated benzenoids, the results indicate that the weighted F indexes were in accordance with their chiral classification (achiral or chiral) over the whole benzenoids dataset. Furthermore, weighted F indexes were superior to previously available descriptors. Benzenoids possess a variety of shapes and can be extended to practically represent any shape in 2D space—our proposed descriptor has thus the potential to be a general method to represent 2D molecular chirality based on the difference between clockwise and anticlockwise sums around a molecule.

where, h is the number of hexagons of each enumerated benzenoid; m is the integer part of the quotient of (2h+1)/3. The lengths of the two bases are h and h-m+1.
In a hexagonal lattice, a benzenoid of size h can be regarded as planar connected polyhexes, called h continuous hexagons here. The proof based on pure graph theory is introduced as follows.
A coordinate system (x, y) in a hexagonal lattice is shown in Figure 1. In the system, every hexagon in the grid can be represented by a unique coordinate. Then a third axis is introduced as shown in Figure 2. Every hexagon still has a unique coordinate. And for every coordinate (x, y, z), it is easy to find that the equation x + y = z is always correct. For example, the coordinates of four grey hexagons in Figure 2 are (0, 0, 0), (1, 1, 2), (1, 2, 3) and (2, 2, 4). from a starting hexagon (x0, y0, z0), z of any hexagon is less than z0+h.
Lemma Lemma Lemma Lemma 2: 2: 2: 2: If h-1 continuous hexagons are added one by one from a starting hexagon (x0, y0, z0) and the coordinate y of one added hexagon is 0, the value of z of any added hexagon is less than z0+h-y0., i.e., z< z0+h-y0.

Proof Proof Proof Proof
For a hexagon (x, y, z), there are six neighbors as above. The coordinates y of two neighbors decrease, and their coordinates are (x+1, y-1, z) and (x, y-1, z-1). It can be found that the coordinates satisfy: zN≤z. When the coordinate of starting hexagon is (x0, y0, z0) and the coordinate y of one added hexagon is 0, at least y0 hexagons satisfying yN= y-1 need be added, that is, at least y0 hexagons are added, but z doesn't increase. In this case, if h-1 continuous hexagons are added from the starting hexagon, based on lemma 1 the coordinate of any added hexagon satisfy: z-z0< h-y0, that is, the value of z of any added hexagon is less than h-y0+ z0., i.e., z<h-y0+ z0.
Lemma Lemma Lemma Lemma 3: 3: 3: 3: Any Any Any Any benzenoid benzenoid benzenoid benzenoid composed composed composed composed of of of of h h h h hexagons hexagons hexagons hexagons can can can can be be be be placed placed placed placed in in in in an an an an equilateral equilateral equilateral equilateral triangular triangular triangular triangular area area area area whose whose whose whose edge edge edge edge consists consists consists consists of of of of h h h h hexagons. hexagons. hexagons. hexagons.

Proof Proof Proof Proof
An equilateral triangular area (the length of any edge is h) on hexagonal lattice is shown in Figure 3 and the coordinate of O is the origin (0, 0, 0). Any benzenoid composed of h hexagons placed on the grid is required to follow two predefined rules: 1) For the coordinate (x, y, z) of any hexagon, there must be x ≥ 0 and y ≥ 0; 2) At least one hexagon is placed on x-axis and one hexagon is on y-axis.
It is easy to find that each hexagon on the line DE satisfies the condition z=h-1. In this case, if z < h can be proved for every hexagon (x, y, z) contained in any benzenoid size of h, lemma 3 is proved.
The coordinate of hexagon A is (0, |OA|, |OA|) and the coordinate of hexagon B is (|OB|, 0, |OB|). When hexagon A is regarded as starting hexagon, the remaining h-1 hexagons were added one by one, and z of any added hexagon is less than h-|OA|+|OA| =h based on lemma 2 (z<h-y0+ z0). Thus, it is proved that z < h. Figure Figure Figure Figure 3 3 3 3 h h h h and and and and h h h h----m m m m+1 +1 +1 +1. . . .

Proof Proof Proof Proof
A benzenoid can be rotated in 2D space, and the different poses require different size of trapezoidal area. As an example of a benzenoid shown in Figure 4, there are six ways to place a benzenoid and all the six trapezoidal areas have the same number of hexagons (h) along the lower base, which is in accordance with Lemma 3. The benzenoid that looks like a clover has the largest value of min(X, Y, Z) of all the benzenoids with a certain number of hexagons. In order to get this benzenoid a hexagon is added to one of the leaves circling around the central hexagon every time.
The procedure of adding hexagons is shown in Figure 6, and herein X = min(X, Y, Z). x