ADNAN LIAQUATFA16-RMT-003Dr. Imran AhmedLHR TP 51782026-04-212018https://repository.cuilahore.edu.pk/123456789/3636In 2017, Anwar, Kosar and Nazir introduced the Gallai simplicial complex ∆Γ(G) associated to a simple graph G. The homology is a powerful tool in algebraic topology. It formalizes the concept of holes related to connected components of a simplicial complex. The Betti numbers are topological invariants that are used to calculate the number of holes of each dimension in a space. In Theorem 4.0.26, we compute the formula for Betti numbers of the Gallai simplicial complex ∆Γ(F2n+1) associated to friendship graph F2n+1 by coding in MATLAB. We define first the Gallai total simplicial complex ∆Γ(G) associated to a simple graph G. In Theorem 5.0.30, we prove that the Gallai total simpli cial complex ∆Γ(G) is connected iff G is connected. In Lemma 5.0.31, we compute f-vector of the Gallai total simplicial complex ∆Γ(L∗ 2n) associated to triangular ladder graph L2n. In Theorem 5.0.32, we find the formula of Betti numbers for ∆Γ(L∗ 2n) by coding in MATLABenDr. Imran AhmedMATHEMATICSFA16Thesis