Cohen-Macaulayness of Total Simplicial Complexes

2023/10/15 by Abbas, Najam Ul, Ahmed, Imran, Kiran, Ayesha
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.09966

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 Δ_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.

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