M.Phil / MS
Permanent URI for this collectionhttps://repository.cuilahore.edu.pk/handle/123456789/52
This collection archives the complete set of theses produced by students of the COMSATS University Islamabad, Lahore Campus.
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Item Characterization of Line Total Simplicial Complexes(Library Information Services, COMSATS University, Lahore Campus, 2018) AYESHA ANDALIB KIRAN,; FA16-RMT-033; Dr. Imran Ahmed; LHR TP 5196In 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 MATLABItem Characterizations of Gallai Total Simplicial Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2018) ADNAN LIAQUAT; FA16-RMT-003; Dr. Imran Ahmed; LHR TP 5178In 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 MATLAB