Neighborhood Complexes of Some Exponential Graphs

In this article, we consider the bipartite graphs $K_2\times K_n$. We first show that the connectedness of $\mathcal{N}(K_{n+1}^{K_n})=0$. Further, we show that $\text{Hom}(K_2\times K_n,K_m)$ is homotopic to $S^{m-2}$, if $2\le m<n$.