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 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 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.Item Properties of Total Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) BISMA ATIQ; CIIT/FA20-BSM-007/LHR; Dr. Imran Ahmed; LHR TP 9913With every graph G (finite and undirected with no loops or multiple lines) there is associated a graph L(G), called the line-graph of G, whose points correspond in a one-to-one manner with the lines of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of G are adjacent. Let G = (V, E) be a graph. The Gallai total graph ΓT (G) of G is the graph, where V (ΓT (G)) = V ∪ E and uv ∈ E(ΓT (G)) if and only if (i) u and v are adjacent vertices in G, or (ii) u is incident to v or v is incident to u in G, or (iii) u and v are adjacent edges in G which do not span a triangle in G. The anti-Gallai total graph ∆T (G) of G is the graph, where V (∆T (G)) = V ∪ E and uv ∈ E(∆T (G)) if and only if (i) u and v are adjacent vertices in G, or (ii) u is incident to v or v is incident to u in G, or (iii) u and v are adjacent edges in G and lie on a same triangle in G. The Gallai middle graph ΓM(G) of a graph G = (V, E) is the graph whose vertex set is V ∪E and two edges ei, ej ∈ E are adjacent in ΓM(G), if they are adjacent edges of G and do not lie on a same triangle in G, or if e = uv ∈ E then e is adjacent to u and v in ΓM(G). The anti-Gallai middle graph ∆M(G) of G is the graph whose vertex set is V ∪E and two edges ei, ej ∈ E are adjacent in ∆M(G) if they are adjacent in G and lie on a same triangle in G, or if e = uv ∈ E then e is adjacent to u and v in ∆M(G).In this paper, we present Eulerian and Hamiltonian properties of Gallai and anti-Gallai middle graphs. In this paper, we investigate Total Graph, Gallai and anti-Gallai Total Graphs and Middle graph.Item Characterization of Total Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) SANIA MASOOD; CIIT/FA20-BSM-016/LHR; Dr. Imran Ahmed; LHR TP 9914This thesis delves into the characterization and properties of regular total graphs and their chromatic numbers. A regular total graph is a unique graph structure where each vertex has the same degree, and the total graph, denoted as T(G), incorporates both the vertices and edges of a given graph G as vertices, with edges in T(G) representing adjacency or incidence relationships in G. The primary aim of this research is to elucidate the structural characteristics and chromatic properties of these regular total graphs. We consider “ordinary” graphs; that is, finite un-directed graphs with no loops or multiple edges. The total graph T(G) of a graph G is that graph whose vertex set is V(G)U E(G) and in which two vertices are adjacent if and only if they are adjacent or incident in G. A characterization of regular total graphs as well as some other properties of total graphs have been considered before. In this article we consider nonregular graphs and yield a method which enables us actually to determine whether or not they are total. Let G be an ordinary graph (finite, undirected, with no loops or multiple lines). Besides the chromatic number X(G) and line chromatic number X 0 (G), there is associated with G another positive integer X 00 (G), called the total chromatic number of G, which is the minimal number of colours required for colouring the elements (points and lines) of G such that no two elements which are either adjacent or incident have the same colour. The chromatic number, a fundamental parameter in graph theory, represents the minimum number of colors required to color the vertices of a graph such that no two adjacent vertices share the same color. This research aims to depend the understanding of how chromatic properties manifest in complete graphs and regular graphs. Regular graphs, the relationship between vertex degree and chromatic number is scrutinized, with new bounds and exact values established for specific classes of regular graphs. For complete graphs, the study reaffirms that the chromatic number is equal to the number of vertices, providing a clear and concise analysis of this well-known result. Closed loop cycle paths, also known as cycles, are fundamental structures in graphs where a sequence of vertices is connected by edges forming a loop with no repeats except for the starting and ending vertex. Understanding these cy- cles is crucial for various applications in computer science, network theory, and optimization problems. The thesis then explores different types of cycles, such as Hamiltonian and Eulerian cycles, and their significance in both directed and undirected graphs. Automorphisms and isomorphisms are fundamental concepts in graph theory, playing a crucial role in understanding the symmetry and equivalence of graph- s. Automorphisms, which are graph isomorphisms from a graph to itself, are explored in detail to understand the symmetries within a graph. Isomorphisms, which are bijections between vertex sets of two graphs that preserve adjacency, are analyzed to determine graph equivalence.Item Connectivity of Total Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rimsha Imran; CIIT/FA20-BSM-033/LHR; Dr. Imran Ahmed; LHR TP 9915We associate with a graph (finite, undirected, without loops and multiple lines) a graph T(G), called the total graph of G. This new graph has the prop- erty that a one-to-one correspondence can be established between its points and the elements (points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. Connectivity is a basic concept of graph theory. It defines whether a graph is connected or disconnected. Without connectivity, it is not possible to traverse a graph from one vertex to another vertex. A graph is said to be connected graph if there is a path between every pair of vertex. From every vertex to any other vertex there must be some path to traverse. This is called the connectivity of a graph. In graph theory, an automorphism of a graph G is a permutation of the vertices that preserves the adjacency structure of the graph. In simpler terms, it’s a re-labeling of the vertices such that the overall shape or structure of the graph remains unchanged. Formally, let G=(V,E) be a graph with vertex set V and edge set E. An automorphism of G is a bijective function f:V → V such that for any two vertices u,v in V ,u is adjacent to v if and only if f(u) is adjacent to f(v).