شاہ ولی اللہ کی تعلیمات کی روشنی میں خانگی نظام: ایک تحقیقی جائزہ

Every Human being needs family system to live with dignity. There is joint family system, and separate family system in practice in Muslim Societies. Shah Waliullah gave concept of ideal family system. The ingredients of ideal family system of Shah Waliullah are Nikāḥ, Siblings, Property, Love and Society. People faced many problems for living respectfully, and born grave losses due to war. Shah Waliullah not only taught us religious teachings but also guided us how to make our socio- economic system strong as well as family system. In this article, his teachings and thoughts about family system have been discussed.

Topological Invariants of Graphs

Chemical graph theory provides useful tools such as molecular descriptors to develop strong intrinsic relationship between the physicochemical features of chemical compounds and their molecular graphs. The study of molecular descriptors provides a theoretical basis for the fabrication of chemical materials and is helpful in making up for the lack of chemical experiments. There are two prominent types of molecular descriptors; the topological indices and counting polynomials. Further, topological indices can be categorized in two major classes: one class is based on degree and the other class is based on distance. In this thesis, we present the study of certain topological indices belonging to degree- and distance-based classes and counting polynomials for some well-known nanostructures. We also present a comparative study between different topological indices belonging to degree- and distance-based classes for general graphs. In addition, we study the para-line transformation of graphs and obtain the general expressions of certain topological indices for this transformation. We achieve the lower and upper bounds of certain distance-based topological indices for the para-line transformation of graphs.