توہین رسالت، خدشات، اعتراضات اور اس کے جوابات

The Holy Quran is that last scripture of Allah Almighty that has been revealed for the guidance of the human beings. Prior of its revelation the mankind was indulged in the darkness of illiteracy, ignorance, barbarism, lawlessness and idolatry. But the Quran revealed them the wealth of knowledge and in the time the followers of this Great Book became mentors of the whole Ummah. In Arabia, mankind had lost their dignity before the Quran was sent. They lacked national unity. They considered it to be a binding of themselves to Quarrel with one another. The past religious teaching had been altered. The life of the poor had been made tough by the local Arab Tribal leaders. Allah almighty Showered by sending the Holy Quran in such circumstances.   The Quran is such a code of life which is a sure Source of success if lead our lives according to its Teachings. The Root Cause of our Problems is our lack of knowledge about the Quranic Teachings. It is the need of Hour that the Teachings of the Quran should be extended so that the Muslim Youth should be able to build their character according to the Quranic principles.

Properties of Graphs With H-Covering and Prescribed H-Weights

Let vertex and edge sets of graph G are denoted by V (G) and E(G), respectively. An edge-covering of G is a family of di erent subgraphs H1;H2; : : : ;Hk such that each edge of E(G) belongs to at least one of the subgraphs Hj , 1 j k. Then it is said that G admits an (H1;H2; : : : ;Hk)-(edge)covering. If every Hj is isomorphic to a given graph H, then G admits an H-covering. For axed graph H, a total labeling : V (G) [ E(G) ! f1; 2; : : : ; jV (G)j + jE(G)jg is said to be H-magic if all subgraphs of G isomorphic to H have the same weight. One can ask for di erent properties of a total labeling. The total labeling is said to be antimagic if the weights of subgraphs isomorphic to H are pairwise distinct. Further restriction on the weights of subgraphs provides (a; d)-H-antimagic labelings where the weights of subgraphs form an arithmetic progression with di erence d and rst element a. If graph G is a 2-connected plane graph then the H-antimagic labeling is equiva- lent to d-antimagic labeling of type (1; 1; 0), where weights of all faces form an arith- metic sequence having a common di erence d and the weight of a face under a labeling of type (1; 1; 0) is the sum of labels carried by the edges and vertices on its boundary. In therst part of the thesis we will study the notions, notations and de nitions about graphs and labeling of graphs. In the second part of the thesis, we have three chapters on newly obtained results. In the chapters, we examine the existence of Hk 2 -supermagic labelings for graphs Gk 2 obtained from two isomorphic graphs G and G0 by joining every couple of corre- sponding vertices v 2 V (G) and v0 2 V (G0) by a path of length k + 1. We show that graphs Gk(w), obtained from a graph G by joining all vertices in G to a vertex w by paths of length k + 1, keep super H-antimagic properties of the graph G. We also examine the existence of the (H G2)-supermagic labelings of Cartesian product G1 G2, where G1 admits an H-covering and G2 is a graph of even order. Addition- ally, we show that if a graph G admits a (super) (a; 1)-tree-antimagic labeling then the disjoint union of multiple copies of the graph G keeps the same property.