Analytical forms of the coefficients of the factored characteristic polynomials for zigzag SWCNT graphs

Abstract

Factors of characteristic polynomials (CP) of zigzag single walled carbon nanotube (SWCNT) graphs have been constructed in generalized forms using rotational symmetry with respect to the tube axis. The CP coefficients in the factors have been expressed in analytical forms involving derivatives of different powers of pj, where pj=(1+ωj)(1+ωj*), ωj=0,1,2,…,(R−1), are the R-th roots of unity and R is the number of fused hexagonal rings per belt in the SWCNT graph. Some important results regarding total π-electron energy per electron, HOMO-LUMO energy gap and number of Kekulé structures for the SWCNTs have been derived from an analysis of the CPs and graph eigenspectra.