Cohen-Macaulayness of Total Simplicial Complexes

Abstract

We first construct the total simplicial complex (TSC) of a finite simple graph $G$ in order to generalize the total graph $T(G)$. We show that $\Delta_T(G)$ is not Cohen-Macaulay (CM) in general. For a connected graph $G$, we prove that the TSC is Buchsbaum. We demonstrate that the vanishing of first homology group of TSC associated to a connected graph $G$ is both a necessary and sufficient condition for it to be CM. We find the primary decomposition of the TSC associated to a family of friendship graphs $F_{5n+1}$ and prove it to be CM.