On Diameter of Nanotubical Fullerene Graphs

Fullerene graphs are 3-connected 3-regular planar graphs with only pentagonal and hexagonal faces ; if no two pentagonal faces are incident we call it an isolated pentagon fullerene. We show that the diameter of an isolated pentagon fullerene graph $G$ of order $n$ is at most $(n+12)/9$. Moreover, if $G$ is not a $(9, 0)$-nanotube its diameter is at most $n/10 + 3$. Additionally, we show that the diameter of a $(p, q)$-nanotube is essentially $n/(p+q)$.

[1]  Vesna Andova,et al.  Diameter of Fullerene Graphs with Full Icosahedral Symmetry , 2013 .

[2]  Pierre Hansen,et al.  Facts and Conjectures about Fullerene Graphs: Leapfrog, Cylinder and Ramanujan Fullerenes , 2001 .

[3]  T. Došlić.  Bipartivity of fullerene graphs and fullerene stability , 2005 .

[4]  Michael Goldberg,et al.  A Class of Multi-Symmetric Polyhedra , 1937 .

[5]  S. Iijima.  Helical microtubules of graphitic carbon , 1991, Nature.

[6]  Tomislav Došlić.  Saturation number of fullerene graphs , 2008 .

[7]  Tamás Réti,et al.  Spectral Properties of Fullerene Graphs , 2011 .

[8]  Patrick W. Fowler,et al.  A census of nanotube caps , 1999 .

[9]  Douglas J. Klein,et al.  Theorems for carbon cages , 1992 .

[10]  Joseph Malkevitch,et al.  Geometrical and combinatorial questions about fullerenes , 1998, Discrete Mathematical Chemistry.

[11]  Branko Grünbaum,et al.  The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra , 1963, Canadian Journal of Mathematics.

[12]  Riste Škrekovski,et al.  On the diameter and some related invariants of fullerene graphs , 2012 .

[13]  Tomislav Doslic,et al.  Computing the bipartite edge frustration of fullerene graphs , 2007, Discret. Appl. Math..

[14]  Craig E. Larson,et al.  Graph-Theoretic Independence as a Predictor of Fullerene Stability , 2003 .

[15]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[16]  Empirical Study of Diameters of Fullerene Graphs , 2011 .

[17]  Fengshan Bai,et al.  A Note on Fowler-Manolopoulos Predictor of Fullerene Stability , 2010 .

[18]  Gunnar Brinkmann,et al.  A constructive enumeration of nanotube cap , 2002, Discret. Appl. Math..

[19]  Gunnar Brinkmann,et al.  Pentagon-hexagon-patches with short boundaries , 2003, Eur. J. Comb..

[20]  D. Manolopoulos,et al.  An Atlas of Fullerenes , 1995 .

[21]  Patrick W. Fowler,et al.  Pentagon adjacency as a determinant of fullerene stability , 1999 .

[22]  Tomislav Došlić,et al.  On Some Structural Properties of Fullerene Graphs , 2002 .