  In this article, local limit theorems for sequences of simple random walks on
graphs are established. The results formulated are motivated by a variety of
random graph models, and explanations are provided as to how they apply to
supercritical percolation clusters, graph trees converging to the continuum
random tree and the homogenisation problem for nested fractals. A subsequential
local limit theorem for the simple random walks on generalised Sierpinski
carpet graphs is also presented.
