You may want to consider upgrading the power.law.fit function in igraph to profit from developments documented in
which allow in-function assessment of the minimal degree value for the estimation of alpha. As documented in the article, an implementation is available - also for R - in
http://www.santafe.edu/~aaronc/powerlaws/ (function plfit.r). The results seem in line with the current igraph implementation for the graph I am working on with the added advantage of having the minimum degree estimated automatically!