Department of Mathematics
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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).Item Properties of SSC of Wheel Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2023) Faisal Rehman FA20-BSM-064; Dr. Imran Ahmed; LHR TP 9901Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of the thesis is the Properties of SSC of Wheel Graphs, the emphasis being placed on homology, connectivity degree, Cohen-Macaulayness, and Vertex Decomposition. Let ∆ be a pure simplicial complex on n vertices having dimension d and codimension c = n−d −1 in the simplex. Terai and Yoshida proved that if the number of facets of ∆ is at leastItem Connectivity of Total Graphs(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rimsha Imran; FA20-BSM-033; 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).