Browsing by Author "Faiz Farid"

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    Distance based topological indices of connected graphs
    (UMT Lahore, 2024-10-21) Faiz Farid
    In the modern era, mathematical modeling consists of graph theoretic parameters or invariants are applied to solve the problems existing in various physical science disciplines like computer sciences, physics and chemistry. Topological indices (TIs) are graph invariants frequently used to identify the molecular graph’s different physicochemical and structural properties. Wiener index is the first distance-based TI that is used to compute the boiling point of the paraffin. Various distance based TIs such as degree distance (DD) and the Gutman (GM) index are used to get different physicochemical and structural properties of the molecular graphs. Graph operations are used to develop more graphical structures. Four special graphs derived from a given graph G are defined by Yan et al., first operation or subdivision graph (S(G)), second operation or triangle parallel graph (R(G)), third operation or line superposition graph (Q(G)), and four operations or total graph T (G), can be used to create new graphs. Later on, Eliasi and Taeri defined F-sum graphs using the Cartesian product on the graph F (G1) and G2, where F (G1) is obtained by applying the operation F ∈ {S, R, Q, T}. These F-sum graphs create hexagonal chains, which are isomorphic to various chemical compounds. They also computed the Wiener index (WI) of these F -sum graphs such as S-sum (G1 +S G2), R-sum G1 +R G2, Q-sumG1 +Q G2 and T-sum (G1 +T G2) graphs. One of the most important distance based TI is the degree distance index DD index. The UBs of the DDIs for all the F -sum graphs were computed by Minigqiang. The exact formulae of the (DD) index for the four different types of the sum-graphs in terms of various indices of their factor graphs are computed by the author. Moreover, the obtained results are illustrated with the help of examples and a comparison is also presented between the exact and bounded values for the particular sum graphs and their certain values. Gutman GM index of a connected graph is also a degree - distance based topological index. In the extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. Pattabiraman computed the UBs of the Gutman indices for the four sum graphs. The exact values of F -sum graphs are determined by the author in tems of other TIs of actual graph G. Moreover, the obtained results are illustrated with the help of examples and a comparison is also presented between the exact and bounded values for the particular sum graphs and their certain values. The relation between the connection numbers of the old and new vertices of derived graphs and the degree or the connection numbers of the vertices of G are also established by the author. These relations have opened new horizons and are a great source for using the connection numbers of the derived graphs. Recently, two TIs are introduced as the connection distance CD index and the Gutman connection GC index. The connection distance CD index is the latest developed TI that is defined as the sum of all the products of distances between pairs of vertices with the sum of their respective connection numbers while GC index is defined as the sum of all the products of distances between pair of vertices with the product of their respective connection numbers. In this thesis, we computed (CD) and (GC) indices of the different derived graphs (subdivision graph S(G), vertex-semitotal graph R(G), edge-semitotal graph Q(G) and total graph T (G)) obtained from the graph G under various operations of subdivision in the form of TIs of the basic graphs including some other algebraic expressions. Two new TIs reciprocal distance connection sum (RDCS) and reciprocal distance connection product RDCP are introduced by the author. First TI RDCS index” of a graph G is defined as the sum of the product of the reciprocal of the distance of two nodes with the sum of the connection numbers of both the nodes while the second TI RDCP index” of a connected and simple graph G is defined as is the sum of the product of the reciprocal of the distance of two nodes with the product of the connection numbers of both the nodes. Four derived graph operations are used to enhance graph structure. In this thesis, RDCS and RDCP of these four operations generated by the graph G in various types of subdivision in a manner of other indices of the parent graphs, involving various other algebraic expressions.
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    Soft zorn's lemma
    (University of Management & Technology, 2017) Faiz Farid
    The present research work is about ets and Soft Matrices. The research shows that set of soft matrices and soft sets over the same universal set obey