J Math Chem (2009) 45:513–524 DOI 10.1007/s10910-008-9424-1 ORIGINAL PAPER Tubercular fulleroids A. E. Vizitiu · Cs. L. Nagy · M. Stefu · G. Katona · M. V. Diudea · B. Parv · D. Vukiˇ cevi´ c Published online: 7 August 2008 © Springer Science+Business Media, LLC 2008 Abstract New tubercular fulleroids are built up by using the three classical compos- ite map operations: tripling (leapfrog Le), quadrupling (chamfering Q) and septupling (capra Ca) on the trivalent Platonic solids. These transforms belong to the tetrahedral, octahedral and icosahedral symmetry groups and show interesting mathematical and (possible) physico-chemical properties. Keywords Fullerenes · Map operations · Leapfrog · Chamfering · Capra · Opening operation · Kekulé valence structure · Kekulé structure count 1 Introduction A map M is a combinatorial representation of a closed surface [1]. Several transfor- mations (i.e., operations) on maps are known and used for various purposes. Let us denote in a map: v, number of vertices; e, number of edges; f , number of faces; and d , vertex degree. A subscript “0” will mark the corresponding parameters in the parent map. Recall some basic relations in a map [2]:  d v d = 2e (1) A. E. Vizitiu · Cs. L. Nagy · M. Stefu · G. Katona · M. V. Diudea (B ) Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, 400028 Cluj, Romania e-mail: diudea@chem.ubbcluj.ro B. Parv Faculty of Mathematics and Computer Science, Babes-Bolyai University, 400028 Cluj, Romania D. Vukiˇ cevi´ c Department of Mathematics, University of Split, Nikole Tesle 12, 21000 Split, Croatia 123