Department of Mathematics
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Item Topological Indices of Fuzzy Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Arslan Kaleem; CIIT/FA19-RMT-108/LHR; Dr. Imran Ahmed; LHR TP 7405In this thesis, We study the topological indices of fuzzy graphs. A topological index is a numerical number for a structural graph of a molecule. It plays an important role in mathe- matical chemistry. We intend to study topological indices of graphs and fuzzy graphs. We establish the relation between first Gourava index and first Gourava co-index. We define some new topological indices of fuzzy graphs. Then we determine the relation between first Zagreb index(respectively second Zagreb index) of fuzzy graph and first Zabreb in- dex(respectively second Zagreb index) of its complete fuzzy graph.Item Fundamental Group and Singular Homology(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) DANIAL HUSSAIN; CIIT/FA19-RMT-008/LHR; Dr. Imran Ahmed; LHR TP 7442Algebraic topology is a very fascinating and beautiful subject. In this area of math- ematics, we study shapes, where we are interested in what is maintained when we continuously deform shapes. The main objective of my thesis is to introduce the basic notions of algebraic topology, such as fundamental group π1 and singular homology group H1(these groups are isomorphic for homotopically equivalent topo- logical spaces). My thesis work is self-contained and explanatory. In 2nd chapter, I have discussed the “fundamental group, its properties, and some very important theorems like the Borsuk-Ulam theorem, Brouwer fixed-point theorem, and Van- Kampen’s theorem”. While “singular homology, its basic properties, homology of a point, induced map between chain complexes, pushforward of homology, homotopy invariance, relative homology, long exact sequence property, induced maps, exci- sion theorem, homology of a quotient, and relation between H1 and π1” have been discussed in chapter 3.Item Classifications of Determinantal Varieties(Library Information Services, COMSATS University, Lahore Campus, 2019) Sana Zahoor; , FA17-RMT-054; Dr. Imran Ahmed; LHR TP 5675Let Mns be a space of matrices of order n×s with complex enteries. The determinantal singularities are defined by vanishing of all the minors of a certain size of an n ×s matrix α : Ck → Mns. The class of determinantal singularities is a more general class than the class of complete intersection singularities. The G-finite deteminacy gives a special class of determinantal varieties known as essentially isolated determinantal singularities (EIDS), firstly introduced by Ebeling and Gusein-Zade in 2009. We give a computational method to find linear forms of EIDS by programming in CItem 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 MATLABItem Algebraic Characterization of Fuzzy Graphs(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Mubeen Naz; SP23-RMT-017; Dr. Imran Ahmed; LHR TP 9579The aim of the thesis is to investigate the algebraic properties of fuzzy graphs with par ticular focus placed on three main topics which are the square of a graph, minimal free resolutions of modules, and fuzzy edge ideals. We will examine the fuzzy graphs relation ships to fundamental algebraic concepts like rings, ideals, and edge ideals by examining their structural and algebraic characteristics. The study focuses on how graph-theoretical concepts and algebraic methods work together. It gives information about minimal free resolutions and the behavior of fuzzy square graph and edge ideals of fuzzy square graph. These results advance the theoretical underpinnings and practical uses of fuzzy commuta tive algebraItem A Note on Determinantal Varieties(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Ayesha Bashir; FA22-RMT-011; Dr. Imran Ahmed; LHR TP 9340The disappearance of all minors of a specific size in a r ×s matrix indicates determinantal varieties. “Essential isolated determinantal singularities (EIDS)” are a special class of de terminantal variations that were created by Ebeling and Zade in 2009. Pereira demonstrated that, for all 1 ≤ k ≤ min{r,s} the determinantal variety Z k = A−1Item Cohen-Macaulayness in Codimension for Line Simplicial Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Nasim Aftab; SP19-RMT-014; LHR TP 7350; Dr. Imran AhmedCohen-Macaulay complexes were first proposed in the mid-1970s. As the primary devel- oper of these inventions, Stanley has made major contributions over a long period of time. The idea of CMt simplicial complexes was itroduced by Nahandi, Yassemi, and Haghighi in 2012. We show that cycle graph having j vertices is unmixed for odd j 3 and it is not unmixed for even j 6 and prism graph having Jj vertices with J 3, j 2 is unmixed for odd J 3 and it is not unmixed for even J 6. Furthermore, we prove that the line simplicial complexes associated to cycle graph having j vertices are unmixed for odd j 3 and it is not unmixed for even j 6 and line simplicial complexes associated to prism graph YJ,2 with J 3 is unmixed. Also, we show that the line simplicial complexes associated to ladder graph having 2n vertices is Cohen-Macaulay and consequently it is CMt, for t 0. We provide example that DL(C5) is not CM0 but it is CMt, for t 1 .Item Classification of Fuzzy Varieties(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2020) Imtiaz Tariq; FA19-RMT-091; LHR TP 7618; Dr. Imran AhmedIn this thesis, we study the fuzzy algebraic varieties. The goal of discussing fuzzy algebraic varieties is to provide some significance to the fuzzy commutative ring theory that has been created thus far and to show the way forward for additional research. We apply fuzzy commutative ring theory to a natural application field, specifically the solution of nonlinear systems of fuzzy singleton intersection equations. It is demonstrated by Mordeson in 1993 that every fuzzy algebraic variety V with 0 ∈ Im(V ) could be represented uniquely as union of a finite number of irreducible algebraic varieties, none of which are contained in union of the others.Item Computational aspects of SSC of Wheel Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rimsha Amber; CIIT/FA20-BSM-058/LHR; Dr. Imran Ahmed; LHR TP 9923Simplicial complexes represent crucial mathematical structures with broad applications. Stanley-Reisner ideals encode combinatorial information about these complexes, bridg- ing combinatorial geometry and commutative algebra. Simplicial complexes, comprised of simplices, capture complex topological and combinatorial relationships, serving as founda- tional tools in numerous mathematical disciplines. Shellable complexes, a significant sub- class, allow systematic decomposition into ”shells,” revealing insights into their topological and homological properties. Studying the spanning simplicial complexes of wheel graphs elucidates the connection between graph theory and algebraic topology. Investigation into the shellability of line simplicial complexes sheds light on their combinatorial intricacies, contributing to a deeper understanding of their geometric and algebraic nature. Through these explorations, we uncover the rich interplay between combinatorial geometry, alge- braic structures, and topological properties inherent in simplicial complexes, facilitating broader comprehension and application across mathematical domains.