Naslov
Lego-like spheres and tori

Autori
Deza, Michel ; Dutour Sikirić, Mathieu

Izvornik
Journal of mathematical chemistry (0259-9791) 55 (2017), 3; 752-798

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Plane graph ; Tori ; Combinatorial enumeration ; Face decomposition

Sažetak
Given a connected surface F2 with Euler characteristic χ and three integers b>a≥1<k, an ({;a, b}; ; k)-F2 is a F2-embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6 ; 3)-S2. By pa, pb we denote the number of a-gonal, b-gonal faces. Call an ({;a, b}; ; k)-map lego-admissible if either pbpa, or papb is integer. Call it lego-like if it is either abf-lego map, or afb-lego map, i.e., the face-set is partitioned into min(pa, pb) isomorphic clusters, legos, consisting either one a-gon and f=pbpab-gons, or, respectively, f=papba-gons and one b-gon ; the case f=1 we denote also by ab. Call a ({;a, b}; ; k)-map elliptic, parabolic or hyperbolic if the curvature κb=1+bk−b2 of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic ({;a, b}; ; k)-S2 with (a, b)≠(1, 2). No ({;1, 3}; ; 6)-S2 is lego-admissible. For other 7 families of parabolic ({;a, b}; ; k)-S2, each lego-admissible sphere with pa≤pb is afb and an infinity (by Goldberg–Coxeter operation) of abf-spheres exist. The number of hyperbolic abf({;a, b}; ; k)-S2 with (a, b)≠(1, 3) is finite. Such afb-spheres with a≥3 have (a, k)=(3, 4), (3, 5), (4, 3), (5, 3) or (3, 3) ; their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible ({;a, b}; ; k)-S2 with pb=2≤a is afb. We list, explicitly or by parameters, lego-admissible ({;a, b}; ; k)-maps among: hyperbolic spheres, spheres with a∈{;1, 2};, spheres with pb∈{;2, pa2};, Goldberg–Coxeter’s spheres and ({;a, b}; ; k)-tori. We present extensive computer search of lego-like spheres: 7 parabolic (pb-dependent) families, basic examples of all 5 hyperbolic afb (b-dependent) families with a≥3, and lego-like ({;a, b}; ; 3)-tori.

Izvorni jezik
Engleski

Znanstvena područja
Kemija

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