Random eigenvalues of nanotubes

Abstract The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in graphene and carbon (p,q)-nanotubes . The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> on their respective graph representations. Considering that a carbon ( p , q )-nanotube can be thought of as a graphene sheet rolled up in a way determined by the chiral vector ( p , q ), our findings are based on a previous study on random eigenvalues of both the hexagonal and triangular lattices. This study reveals that for any given chiral vector ( p , q ), the sequence of counts of closed paths forms a moment sequence derived from a function of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including the probability density function and the moment generating function, are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the circumference of a ( p , q )-nanotube approaches infinity, i.e. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> , the ( p , q )-nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length k , indicating weak convergence of the underlying distributions.