المقاصد عند الإمام الشاطبي

Scholars of the Usul -early and recent- have paid utmost importance to the Maqâsid al-Shariah or the Objectives of Shariah and tried to substantiate, associate and link them with its sources, Qur’an and Sunnah, then to derive from them the Legislator’s or the Shari‘s purpose, the revelation’s true purpose and how it benefits the creatures in terms of promoting what is in their interest and shunning away what is harmful. These objectives became the point of interest for the jurists and scholars who toiled to understand the texts and what they connoted and they continue to venture into its depths and extract its treasures. It becomes quite evident from their exegeses of Qur’an and the Prophetic Traditions and elucidations of the juristic principles and issues. Imam al-Shâtibī and Ibn ‘Ashoor are two such scholars in this field who studied the Objectives of Shariah and contributed to this discipline so greatly that they became reference points for those who came after them. The basic principles in the field that were inferred by them are still its pillars. The Objectives of Shariah are divided into four kinds with reference to their benefits: the necessities, needs, improvements and supplements. The necessities are further divided into protection of Deen, protection of self, of intellect, of possessions and of race among others. Prudence demands that all means should be employed to safeguard these necessities. Thus knowing the Objectives of Shariah and their promulgation among the masses is quite important for the society since it leads towards promotion of the beneficial and shunning away the harmful elements in it.

On Irregular Total and Graceful Labelings

A graph labeling is an assignment of integer label to the elements of graph in such a way that some certain conditions are satisfied. If the domain set consists only vertices (or edges) then it is called vertex (or edge) labeling respectively. If the domain set consists vertices and edges then it is called total labeling. A graceful labeling of a (p, q)-graph is an injection f from the set of vertices to the set {1, 2, . . . , q + 1} such that each edge uv is assigned the label |f (u) − f (v)|, the resulting edge labels are distinct. Moreover if f has the property that there exists an integer λ such that for each edge uv either f (u) ≤ λ < f (v) or f (v) ≤ λ < f (u) then f is called an α-labeling. A one-to-one map taking the vertices on the integers 1, 2, 3, . . . , p with the property that the edge weight (sum of end points labels ) form an arithmetic progression starting from a and having common difference d, is called (a, d)-edge antimagic vertex labeling. A total k-labeling of a graph G is a labeling from the set of vertices and edges to the set {1, 2, . . . , k}. A total k-labeling is defined to be an edge irregular total k-labeling of the graph G if edge weights are different for all pairs of distinct edges. The minimum value of k for which the graph G has an edge irregular total k- labeling is called the total edge irregularity strength of the graph G, denoted by tes(G). In this thesis, we construct an α-labeling of trees from graceful labeling of smaller trees and using a connection between α-labeling and edge antimagic vertex labeling we obtain a super (a, d)-edge antimagic total labeling of trees. Moreover we present new results on the total edge irregularity strength.