Graphene, when deposited on a supporting substrate with a step edge, may be deformed in the presence of the step edges of the substrate. In this study, we have investigated a spatial variation in the local electronic structure near the step region, by performing first-principles calculations for carpetlike armchair graphene nanoribbons (C-AGNR) and two-dimensional periodic carpetlike graphene sheets (PCGS). Our results indicate no practical difference in the local density of states (LDOS) between those of flat and step regions. Interestingly, however, the PCGS shows a remarkable variation in the LDOS with an external electric field (E-field). Furthermore, we also discuss the dependence of the direction and the magnitude of the applied E-field on the spatial variation in the LDOS. (C) 2014 Published by Elsevier B.V.