A fusene is a simple planar 2-connected graph embedded in the plane with all vertices of degree 2 or 3, all bounded faces (not necessarily regular) hexagons, and all vertices not in the boundary of the outer face of degree 3 (Brinkmann et al. 2002).
Fusenes that are a subgraph of the regular hexagonal lattice are called benzenoids.
Fusenes are perfect.
Let the number of internal vertices of a polyhex be denoted . Then catafusenes (or catacondensed fusenes) have (and are therefore also called "tree-like"), and perifusenes (or pericondensed fusenes) have . The numbers of catafusenes composed of polyhexes are sometimes called Harary-Read numbers, and have the impressive generating function
(OEIS A002216; Harary and Read 1970, Cyvin et al. 1993).
Polyhexes may also be classified on the basis of being geometrically planar (called nonhelicenic) or geometrically nonplanar (called helicenic). Fusenes include the helicenes.
The following table gives the numbers of -hexagon fusenes (Brinkmann et al. 2002, 2003) catafusenes (Harary and Read 1970, Beinecke and Pippert 1974, Knop et al. 1984, Cyvin et al. 1993), catafusenes, planar catafusenes, and simple catafusenes.
fusenes catafusenes planar catafusenes simpl. catafusenes Sloane A108070 A002216 A038142 A018190 1 1 1 1 1 2 1 1 1 1 3 3 2 2 3 4 7 5 5 7 5 22 12 12 22 6 82 37 36 81 7 339 123 118 331 8 1505 446 411 1435 9 7036 1689 1489 6505 10 33836 6693 5572 30086 11 166246 27034 141229 12 829987 111630 669584 13 4197273 467262 3198256 14 21456444 1981353 15367577 15 110716585 8487400 74207910 16 576027737 36695369 359863778 See alsoBenzenoid,
Fullerene,
Polyhex Explore with Wolfram|Alpha ReferencesBeineke, L. W. and Pippert, R. E. "On the Enumeration of Planar Trees of Hexagons." Glasgow Math. J. 15, 131-147, 1974.Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Constructive Enumeration of Fusenes and Benzenoids." J. Algorithms. 45, 155-166, 2002.Brinkmann, G.; Caporossi, G.; and Hansen, P. "A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons." J. Chem. Inf. Comput. Sci. 43, 842-851, 2003.Cyvin, S. J.; Brunvoll, J.; Xiaofeng, G.; and Fuji, Z. "Number of Perifusenes with One Internal Vertex." Rev. Roumaine Chem. 38, 65-77, 1993.Harary, F. and Read, R. C. "The Enumeration of Tree-Like Polyhexes." Proc. Edinburgh Math. Soc. 17, 1-13, 1970.Knop, J. V.; Szymanski, K.; Jeričević, Ž.; and Trinajstić, N. "On the Total Number of Polyhexes." Match: Commun. Math. Chem., No. 16, 119-134, Aug. 1984.Sloane, N. J. A. Sequences A002216/M1426, A018190, A038142, and A108070 in "The On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|AlphaFusene Cite this as:Weisstein, Eric W. "Fusene." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fusene.html
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