Condensed polycyclic aromatic (benzendd) hydrocarbons (polyhexes or PAH’s) play a major role in chemistry, and their carcinogenicity has invested them with special interest. They were classified around 1880 1900 when their structures were elucidated into cato-condensed and pericondensed polyhexes. The former (also called catafusenes) are considered to be devoid of carbon atoms common to three benzenoid rings; the latter (perifusenes) do have such carbon atoms. Balaban and Harary  proposed a new definition based on the dualist graph of such polyhexes. The dualist graph has as vertices the centres of benzenoid rings; the edges of the dualist graph connect vertices corresponding to condensed benzenoid rings, i.e. rings sharing a p w of adjacent carbon atoms. Unlike the dual graphs familiar to graph theorists, and unlike graphs in general, in dualist graphs bond angles are important. Polyhexes whose dualist graphs are acyclic are defined as catafusenes; those whose dualist graphs contain three-membered rings (possibly in addition to acyclic branches) are defined as perifusenes; those whose dualist graphs contain larger rings which are not perimeters of triangulated arrays (possibly in addition to three-membered rings and acyclic branches) are defined as coronafusenes
(or after Dias  aa circulenes). Examples are shown in the fonntilas 30-45. According to this ddinition, all catafhsenes with the same number n of benzenoid rings are isomeric and have the formula C4n+2H3n-H*
Nonbranched catafusenes with n hexagons were enumerated by Balaban and Harary ; for odd n there are
(3“-* + 4 + l) /4 isomers
and for even n there are
(3"~* + 2 3“^ ~ ' + l) /4 isomers.