Algebraic properties of the path complexes of cycles
Abstract: Let $G$ be a simple graph and $\Delta_t \left(G\right)$ be a simplicial complex whose facets correspond to the paths of length $t$ $(t\geq2)$ in $G$. It is shown that $\Delta_t \left(C_n\right)$ is matroid, vertex decomposable, shellable and Cohen-Macaulay if and only if $n=t$ or $n=t+1$,...