The author obtains explicit expressions for the expected value of the Merrifield-Simmons index of a random phenylene chain and a random hexagon chain, respectively. The author also computes the corresponding entropy constants and obtains the maximum and minimum values in both random systems, respectively.
Let
Phenylenes are a class of conjugated hydrocarbons composed of six- and four-membered rings, where the six-membered rings (hexagons) are adjacent only to four-membered rings, and every four-membered ring is adjacent to a pair of nonadjacent hexagons. If each six-membered ring of a phenylene is adjacent only to two four-membered rings, we say that is a phenylene chain. Due to their aromatic and antiaromatic rings, phenylenes exhibit unique physicochemical properties. In Figure
Phenylene chains.
The three types of local arrangements in phenylene chains.
We assume that the probability
By eliminating, “squeezing out,” the squares from a phenylene, a catacondensed hexagonal system (which may be jammed) is obtained, called the hexagonal squeeze of the respective phenylene (see Figure
Phenylenes and the corresponding hexagonal squeezes.
The Wiener index and the number of perfect matchings of a random hexagonal chain
Firstly, let us recall some results in [
Consider
Let
As described above, the phenylene chain
Let
Consider
For a random phenylene chain
By the definition of
By the symmetry,
To solve the recursion equation, we use the method of the generating functions. Set
Solving the equations, we have that
So we have the following result.
If
Specifically, one has that
The following corollary is easily obtained from Theorem
If
It is easy to check that
Similar to the phenylene chain
Let
Consider
For a random hexagon chain
By the symmetry,
Just as above, we set
Solving the equations, we have that
So we have the following result.
If
Specifically, we have that
The following corollary is easily obtained from Theorem
If
It is easy to check that
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundations of China (no. 11401102).