Fusene
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 n_i. Then catafusenes (or catacondensed fusenes) have n_i=0 (and are therefore also called "tree-like"), and perifusenes (or pericondensed fusenes) have n_i=1. The numbers of catafusenes composed of n polyhexes are sometimes called Harary-Read numbers, and have the impressive generating function
| H(x)=1/(24)x^(-2){12+24x+48x^2-24x^3+[(1-x)(1-5x)]^(3/2)-3(5x+3)sqrt((1-x^2)(1-5x^2))-4sqrt((1-x^3)(1-5x^3))}-4 =x+x^2+2x^3+5x^4+12x^5+37x^6+... |
(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 n-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.
See also
Benzenoid, Fullerene, PolyhexExplore with Wolfram|Alpha
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References
Beineke, 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.House of Graphs. Fusene Graphs. C6, The theta graph theta(1,5,5), Elementary 2-Wall, Hex Grid 2,2,2, Hex Grid 3,3,3, and broken wheel 13, spoke positions 1,5,9.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|Alpha
FuseneCite this as:
Weisstein, Eric W. "Fusene." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Fusene.html