  In this communication we present together four distinct techniques for the
study of electronic structure of solids : the tight-binding linear muffin-tin
orbitals (TB-LMTO), the real space and augmented space recursions and the
modified exchange-correlation. Using this we investigate the effect of random
vacancies on the electronic properties of the carbon hexagonal allotrope,
graphene, and the non-hexagonal allotrope, planar T graphene. We have inserted
random vacancies at different concentrations, to simulate disorder in pristine
graphene and planar T graphene sheets. The resulting disorder, both on-site
(diagonal disorder) as well as in the hopping integrals (off-diagonal
disorder), introduces sharp peaks in the vicinity of the Dirac point built up
from localized states for both hexagonal and non-hexagonal structures. These
peaks become resonances with increasing vacancy concentration. We find that in
presence of vacancies, graphene-like linear dispersion appears in planar T
graphene and the cross points form a loop in the first Brillouin zone similar
to buckled T graphene that originates from $\pi$ and $\pi$* bands without
regular hexagonal symmetry. We also calculate the single-particle relaxation
time, $\tau(\vec{q})$ of $\vec{q}$ labeled quantum electronic states which
originates from scattering due to presence of vacancies, causing quantum level
broadening.

Mesoscale and Nanoscale Physics;Disordered Systems and Neural Networks