Characterization of Line Total Simplicial Complexes

Abstract

In 2017, Ahmed and Mahmood introduced the line simplicial complex ∆L(G) associated to a simple graph G. The pth Betti number is the rank of the pth homology group of a topological space and it refers to the number of n-dimensional holes on a topological surface. In Theorem 3.0.15, we give formula for Betti numbers of the line simplicial complex ∆L(Wn+1) associ ated to wheel graph Wn+1 by writing code in MATLAB. We define first the line total simplicial complex ∆T(G) associated to a sim ple graph G. In Theorem 4.0.24, we prove that the line total simplicial complex ∆T(G) is connected iff G is connected. In Lemma 4.0.19, we find f-vector of the line total simplicial complex ∆T(F2n+1) associated to friend ship graph F2n+1. In Theorem 4.0.20, we give formula for Betti numbers of ∆T(F2n+1) by coding in MATLAB