In this paper, we analytically investigate the effects of the two samples of mechanical deformations, i.e., uniaxial and torsional strains on the electronic band structure and the density of states of the single-walled carbon nanotubes (SWCNTs) using the nearest neighbor, the third-nearest neighbor (3-NN-TB) and the fifth-nearest neighbor tight-binding (5-NN-TB) approaches to compare the estimated band gap and the density of states of the three methods. To do this, we not only make use of the previous works but also analytically develop a 5-NN-TB-based model for the investigation of the mentioned types of strains. The purpose of this paper is to employ this model for the investigation of the two types of strain in order to acquire an analytic formula for the band structure under strain and get the advantage of analytic formula (e.g., speed in band structure calculations under strain) in conjunction with the high degrees of accuracy.

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