Algebraic Kekulé structures of benzenoid hydrocarbons

In 1865, August Kekule suggested the structure of benzene according to the structural theory of organic chemistry based on the three assertions—formula of benzene is C6H6, all the hydrogens of benzene are equivalent, and each carbon must have four bonds due to its tetravalency.
He proposed that benzene has a cyclic structure of six carbon atoms attached to one hydrogen atom each, with three alternating pi bonds.

According to the molecular orbital (MO) model, benzene has a planar structure with a regular hexagon of six sp2 hybridized carbons. As shown in Figure 1, each carbon is bonded to three other atoms with C–C–C and H–C–C bond angles of 120°. The C–H bond length is 109 pm, and the C–C bond length is 139 pm which is midway between the single bond length of sp3 hybridized carbons (154 pm) and sp2 hybridized carbons (133 pm).

The formation of carbon-carbon bonds leading to the creation of the carbon chain is the basis of organic chemistry. August Kekulé and Archibald Scott Couper independently developed this idea of carbon chain formation.
Hydrocarbons are the simplest organic compounds composed of carbons and hydrogens. Based on the bond order between carbons, the hydrocarbons are further classified into alkanes, alkenes, and alkynes. 
Alkanes are the simplest hydrocarbons with sp3 hybrid carbon atoms. These sp3...

Like benzene, cyclobutadiene and cyclooctatetraene are cyclic compounds with alternate single and double bonds. However, their chemical behavior differs from benzene, as they are unstable and not aromatic. So, what are the structural characteristics of unsaturated compounds categorized as aromatic?
For the first time, Eric Hückel, a German chemical physicist, derived a set of structural features for a compound to be classified as aromatic. This is now known as Hückel’s rule or the 4n + 2 rule.

Benzene is the simplest aromatic hydrocarbon or arene. The IUPAC names for simple monosubstituted benzene derivatives are derived by adding the substituent's name as a prefix to the parent benzene. For example, halobenzene, where the halogen could be fluoro (F), chloro (Cl), bromo (Br), and iodo (I).

Cycloheptatriene is a neutral monocyclic unsaturated hydrocarbon that consists of an odd number of carbon atoms and an intervening sp3 carbon in the ring. The three double bonds in the ring correspond to 6 π electrons, which is a Huckel number, and therefore satisfies the criteria of 4n + 2 π electrons. However, the intervening sp3 carbon disrupts the continuous overlap of p orbitals. As a result, cycloheptatriene is not aromatic.
Removing one hydrogen from the intervening CH2 group with both...