403 KB29 KB200820252016Tratnik, NikoŽigert Pleteršek, Petraletnik:11številka:2str. 425-435ISSN:1855-3966COBISSID:22678536URN:URN:NBN:SI:doc-QXPSMPH4enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneacube polynomialdistributive latticedistributivna mrežafulerenfullereneKekulé structurekekuléjska strukturakockovni polinommedian graphmedianski grafperfect matchingpopolno prirejanjeresonance graphresonančni grafZhang-Zhang polynomialZhang-Zhangov polinomResonance graphs of fullerenes|A fullerene ?$G$? is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. The resonance graph ?$R(G)$? of ?$G$? reflects the structure of its perfect matchings. The Zhang-Zhang polynomial of a fullerene is a counting polynomial of resonant structures called Clar covers. The cube polynomial is a counting polynomial of induced hypercubes in a graph. In the present paper we show that the resonance graph of every fullerene is bipartite and each connected component has girth 4 or is a path. Also, the equivalence of the Zhang-Zhang polynomial of a fullerene and the cube polynomial of its resonance graph is established. Furthermore, it is shown that every subgraph of the resonance graph isomorphic to a hypercube is an induced subgraph in the resonance graph. For benzenoid systems and tubulenes each connected component of the resonance graph is the covering graph of a distributive lattice; for fullerenes this is not true, as we show with an exampleFuleren ?$G$? je 3-regularen ravninski graf, sestavljen samo iz petkotniških in šestkotniških lic. Resonančni graf ?$R(G)$? grafa ?$G$? odraža strukturo njegovih popolnih prirejanj. Zhang-Zhangov polinom fulerena je preštevalni polinom (rodovna funkcija) resonančnih struktur, imenovanih Clarovi krovi. Kockovni polinom je preštevalni polinom induciranih hiperkock v grafu. V tem članku pokažemo, da je resonančni graf vsakega fulerena dvodelen in da ima vsaka njegova povezana komponenta ožino 4 ali pa je pot. Prav tako dokažemo ekvivalenco Zhang-Zhangovega polinoma fulerena in kockovnega polinoma njegovega resonanč- nega grafa. Pokažemo tudi, da je vsak podgraf resonančnega grafa izomorfen hiperkocki v induciranem podgrafu resonančnega grafa. Pri benzenoidnih sistemih in tubulenih je vsaka povezana komponenta resonančnega grafa krovni graf distributivne mreže; za fulerene to ne drži, kot pokaže primerTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije