Introducing Two Transformations in Fullerene Graphs, Star and Semi-Star

Document Type : Research Paper


University of Applied Science and Technology (UAST), Tehran, IRAN


‎A perfect star packing in a given graph $G$ can be defined as a spanning subgraph of $G$‎, ‎wherein each component is isomorphic to the star graph $K_{1,3}$‎. ‎A perfect star packing of a fullerene graph $G$ is of type $P0$ if all the centers of stars lie on hexagons of $G$‎. ‎Many fullerene graphs arise from smaller fullerene graphs by applying some transformations‎. ‎In this paper‎, ‎we introduce two transformations for fullerene graphs that have the perfect star packing of type $P0$ and examine some characteristics of the graphs obtained from this transformation‎.


Main Subjects

[1] B. Grünbaum and T. S. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Can. J. Math. 15 (1963) 744–751,
[2] D. J. Klein and X. Liu, Theorems for carbon cages, J. Math. Chem. 11 (1992) 199–205,
[3] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995.
[4] R. B. King and M. V. Diudea, The chirality of icosahedral fullerenes: A comparison of the tripling, (leapfrog), quadrupling (chamfering) and septupling (capra) transformations, J. Math. Chem. 39 (2006) 597–604,
[5] T. Došlić, M. Taheri-Dehkordi and G. H. Fath-Tabar, Packing stars in fullerenes, J. Math. Chem. 58 (2020) 2223–2244,
[6] M. V. Diudea, M. Stefu, P. E. John and A. Graovac, Generalized operations on maps, Croat. Chem. Acta 79 (3) (2006) 355–362.
[7] M. Taheri-Dehkordi and G. H. Fath-Tabar, Nice pairs of pentagons in chamfered fullerenes, MATCH Commun. Math. Comput. Chem. 87 (3) (2022) 621–628,
[8] T. Došlić, All pairs of pentagons in leapfrog fullerenes are nice, Mathematics 8 (12) (2020) p. 2135,
[9] G. H. Fath-Tabar, A. R. Ashrafi and D. Stevanović, Spectral properties of fullerenes, J. Comput. Theor. Nanosci. 9 (3) (2012) 327–329,
[10] T. Došlić, M. Taheri-Dehkordi and G. H. Fath-Tabar, Shortest perfect pseudomatchings in fullerene graphs, Appl. Math. Comput. 424 (2022) p. 127026,
[11] L. Shi, The fullerene graphs with a perfect star packing, Ars Math. Contemp. 23 (1) (2023) 1–16,
[12] F. Kardoš, A computer-assisted proof of the Barnette–Goodey conjecture: not only fullerene graphs are Hamiltonian, SIAM J. Discret. Math. 34 (1) (2020) 62–100,
[13] P. W. Fowler, J. E. Cremona and J. I. Steer, Systematics of bonding in non-icosahedral carbon clusters, Theoret. Chim. Acta 73 (1988) 1–26,
[14] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60 Buckminsterfullerene, Nature 318 (1985) 162–163,
[15] J. Petersen, Die Theorie der regularen graphs, Acta Math. 15 (1891) 193–220,