بنجر نہ کبھی کشت تمنا میری ہو گی
بنجر نہ کبھی کشت تمنا میری ہو گی
اک فرد کا مرنا نہیں انسان کا مرنا
سر ہیں تو سرِ دار قلم اور بھی ہوں گے
ہم مرتے رہیں گے مگر ہم اور بھی ہوں گے
کوئی مسیحا نہ ایفائے عہد کو پہنچا
بہت تلاش پسِ قتل عام ہوتی رہی
It is clear from the fact that Allah has revealed the Quran in seven letters. And there are many things hidden in it. These are important to make Quranic readiness easier for people who read, and translate translation of Quranic words, in the contemporary interpretation of the meaning of Wafa'am and Ahmah Mussel, to smooth the path of extravagance and ease for the Umrah period. There are many such sciences that stand on the basis of different types of trees. These verses are explaining the meaningful meaning of Quranic interpretation in the Qur'aan, based on the verses of the Qur'aan, a faqha is a knowledge of a profession in the verses of the Quran, that is, in fact, the interpretation of Salaf is mentioned in the verses, On the basis of them, the Koran receives discrimination and Ejaz, which is mentioned in Koran in case of Kaafir's challenge. وَإنْ کُنْتُمْ فِیْ رَیْبٍ مِّمَّا نَزَّلْنَا عَلٰی عَبْدِنَا فَأتُوْا بِسُوْرَۃٍ مِّنْ مِّثْلِہِ وَادْعُوْا شُھَدَاء کُمْ مِّنْ دُوْنِ اﷲِ إنْ کُنْتُمْ صٰدِقِیْنَ فَاِنْ لَّمْ تَفْعَلُوْا ولن تفعلوا۔ "And if you suspect that this book which we ascend to our servant, it is not ours, so make one Surah like it, call our new ones, except for Allah Take the help you want, if you are truthful then do this work, but you did not do this and could never believe. " In relation to knowledge, other scholars and scholars (interpretation, jurisprudence, problems, beliefs, knowledge, knowledge and virtue of Muslim tradition) will be cleared.
In my thesis some of the techniques of graph transformation and its applications are introduced. Some of the basic graph transformations are edge deletion, edge contraction, vertex splitting, inner dual of a planar graph and vertex deletion. Inner dual of a graph does not maintain the orientation of the edges in the original graph. To keep the orientation, various methods and techniques are used. Here the technique called He-Matrix is used. This is designed for any hexagonal system. Here the orientation of the edges is represented by using edge weights. These weights can be 1, 2 or 3 for any edge depending upon its orientation. When a hexagonal system is rotated through angles which are multiple of 60 degrees a new graph is obtained. Considering the edge weights, the inner dual of these graphs may be different. This implies that the different inner duals can be compared on different basis. In this thesis two different problems are solved related to these inner duals. The first problem is to find the orientation where the minimum spanning tree is the smallest. Different theorems and algorithms related to this topic are given. Also two linear time algorithms are presented. The first one finds the required orientation without computing the minimum spanning tree in any of the direction while the second solves the minimum spanning tree problem in linear time, in any of the given orientation. The second problem related to the Inner dualist is finding the orientation that gives the smallest shortest path between any two given points. Again some theorems are proved and a linear time algorithm is discussed that determines the orientation and finds the required shortest path is presented. The next problem is an application of graph transformation and network flows. Here a postman problem is discussed. This postman is working in an office with some constraints. These constraints are analogous to the working of a router, so the results from this part can be applied to finding faulty routers. The information available is the total number of letters sent and received by each person. The objective is to find some bounds on the total number of letters that the postman has lost. This number can be used to measure the performance of our postman. This can also be used to compare the performance of different routers, and also indicate the routers that are faulty. Two different linear time algorithms are given that solve this problem correctly. Moreover, a few theoretical results are also given. The concept of graph transformation in the area of teaching methodologies is also applied. Group theory is among the most difficult courses in undergraduate mathematics. The difficulty in learning arises from the fact that this course is abstract in nature and no pictorial representation or visualization is used in teaching this subject. Groups can be represented in various ways, and one of the representations is group automata. This is very similar to finite automata. Here a group is first transformed into an automaton, which is also abstract in nature. Representation for automata is available in the form of state diagrams and they can be viewed as a labeled graph. Here, the group axioms can be studied easily, and identity element and inverse of any element can be visualized. Also the closure property and the associative law can be understood with the help of such graphs. After the axioms, some of the fundamental theorems in group theory are proved in this model, and it is shown how group transformations can be used to present a proof using graphs, of otherwise abstract concepts and theorems. One such example is analogous to the technique where an equivalent minimum state deterministic finite automata is found for any given deterministic finite automata.