J Math Chem (2013) 51:1211–1220 DOI 10.1007/s10910-012-0128-1 ORIGINAL PAPER Statistical properties of carbon nanostructures Forrest H. Kaatz · Adhemar Bultheel Received: 9 November 2012 / Accepted: 7 December 2012 / Published online: 20 January 2013 © Springer Science+Business Media New York 2013 Abstract We look at modeling carbon nanostructures from a theoretical graph net- work view, where a graph has atoms at a vertex and links represent bonds. In this way, we can calculate standard statistical mechanics functions (entropy, enthalpy, and free energy) and matrix indices (Wiener index) of finite structures, such as fullerenes and carbon nanotubes. The Euclidean Wiener index (topographical index) is compared with its topological (standard) counterpart. For many of these parameters, the data have power law behavior, especially when plotted versus the number of bonds or the number of atoms. The number of bonds in a carbon nanotube is linear with the length of the nanotube, thus enabling us to calculate the heat of formation of capped (5,5) and (10,10) nanotubes. These properties are determined from atomic coordinates using MATLAB routines. Keywords Fullerenes · Carbon nanotubes · Statistical mechanics · Wiener index · MATLAB 1 Introduction Carbon has many allotropes (diamond, graphite, fullerenes, and carbon nanotubes, (CNTs)) that illustrate the amazing chemical and structural diversity of element num- ber six. We consider the nanosized forms in our calculations; fullerenes and CNTs. Fullerenes were discovered in 1985 [1], carbon nanotubes in 1991 [2], and graphene F. H. Kaatz (B ) Mesalands Community College, 911 South 10th Street, Tucumcari, NM 88401, USA e-mail: fhkaatz@gmail.com A. Bultheel Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium 123