Neighborhood Complexes of Some Exponential Graphs

TL;DR

This paper investigates the topological properties of neighborhood complexes of certain exponential bipartite graphs, revealing their connectedness and homotopy types, which advances understanding of graph complexes in algebraic topology.

Contribution

It demonstrates the connectedness and homotopy equivalences of neighborhood complexes for specific bipartite graphs, providing new insights into their topological structure.

Findings

Neighborhood complex of K_{n+1}^{K_n} is disconnected.

Hom( K_2 × K_n, K_m ) is homotopic to S^{m-2} for 2 ≤ m < n.

Provides topological characterization of exponential bipartite graphs.

Abstract

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 $2\leq m <n$.