  To estimate the lower bound for the chromatic number of a graph $G$, Lov\'asz
associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood
complex and relates the topological connectivity of $\mathcal{N}(G)$ to the
chromatic number of $G$. More generally he proved that the chromatic number of
$G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus
$3$.
  In this article, we consider the graphs of maximal degree at most $3$ and
$4$-regular circulant graphs. We show that each connected component of the
neighborhood complexes of these graphs is homotopy equivalent either to a
point, to a wedge sum of circles, to a wedge sum of $2$-spheres $S^2$, to
$S^3$, to a garland of $2$-spheres $S^2$ or to a connected sum of tori.
