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اداریہ

سیالکوٹ کی تہذیب قدامت کے لحاظ سے پانچ ہزار سال سے بھی پہلے کے آثار ظاہر کرتی ہے۔راجہ شل نے اس تہذیب کو پروان چڑھانے میں اہم کردار ادا کیا۔اس شہر کی تہذیبی روایات اور علمی آثار " مہابھارت" میں بدرجہ اُتم موجود ہیں۔سیالکوٹ کی مٹی بڑی زرخیز اور مردم خیز ہے۔سرزمین سیالکوٹ نے علم وادب وفنون لطیفہ کے میدانوں میں گراں قدر خدمات سرانجام دی ہیں۔سیالکوٹ کی علمی وادبی  اہمیت مسلمہ ہے۔ہر دور میں خواہ وہ ہندو راج ہو ، مغلیہ راج ہویا انگریز راج سیالکوٹ نے ہردور میں علمی وادبی مرکز کے حوالے سے اپنی شناخت قائم رکھی ہے۔یہاں سے بہت سی نامور روحانی اور علمی وادبی شخصیات نے جنم لیا ہےاور بعض نے یہاں کی روحانی اور علمی وادبی شخصیات سے فیض حاصل کیا ہے۔٧٠٠ قبل مسیح سے٦٠٠ قبل مسیح تک یہ اتنا عظیم تعلیمی مرکز تھا۔کہ بنارس کے شہزادے حصول علم کے لیے یہاں آتے تھے۔

اکیسویں صدی عیسویں میں بھی شہرِ اقبال اپنی تہذیبی و ادبی  روایات کی بازیافت کے لیے خاصا سرگرم عمل ہے۔ملا عبدالحکیم سیالکوٹی ،مولانا فیروزالدین،اقبال ،فیض ،مولانا ظفر علی خاں،  ہاشم شاہ،حضرت رائج سیالکوٹی، دلشاد ،منشی میراں بخش جلوہ،محمد الدین فوق ،اثر صہبائی ،سلیم واحد سلیم ،بدری ناتھ سدرشن،جوگندر پال ،غلام الثقلین نقوی ،رجندر سنگھ بیدی،عبدالحمید عرفانی،سرمد صہبائی،خالد نظیر صوفی، ڈاکٹر جاوید اقبال،ساغر جعفری،مولوی ابراہیم میر،آسی ضیائی رامپوری،طفیل ہوشیارپوری،اے ڈی اظہر،حفیظ صدیقی،صابر ظفر،اصغر سودائی اور جابر علی سید دنیائے شعروادب کے اہم ستارے ہیں۔جن کا تعلق سیالکوٹ کی دھرتی کے ساتھ تادمِ حیات رہا ۔موجودہ دور میں بھی خطہ سیالکوٹ علمی وادبی میدان میں مضافاتی دائرے سے نکل کر قومی وبین الاقوامی ادبی دھارےمیں شامل ہونے کے لیے پرتول رہا ہے۔پنجاب لٹریری فورم سیالکوٹ اسی سلسلے میں اہم کردار ادا کررہا ہے۔اس ادبی تحریک کا ثمر اس خطے کی ادبی سرگرمیوں کی نشاة ثانیہ کی...

روايات سيرت كى تحقيق كا حديثى معيار

Seerah is a separate Islamic science from Hadith as their primary sources are different. Although there are some extents where there is over laying between them, but traditionally Seerah has different principles as compared to Hadith. The Scholars of Hadith were very strict in applying their rules whereas the scholars of Seerah were more flexible. The reason is, when academics were dealing with Ahadiths and deducingdivine rulings, they wanted to make sure they were founding the rulings on Ahadiths that were authentic and sound. So that is why they applied very stringent rules to accept Ahadith. However, when it came to Seerah, they were more flexible in their rules, because they study this as history of The Prophet PBUH which does not touch the Sharia rulings. So, we find that writers of Seerah would accept narrations, they would not usually accept if they were dealing with Ahadith. This practice with Seerah narrations was followed by our early scholars. But recently, there is a new movement among some of our researchers that they wanted to apply the rules of Ahadith on Seerah. We do not agree this approach and in this article, we have had a humble effort to compile a set of rules for acceptance of Seerah narrations.

Hamiltonian Properties of Directed Toeplitz Graphs

To determine whether or not a given graph has a hamiltonian cycle is much harder than deciding whether it is Eulerian, and no algorithmically useful characterization of hamiltonian graphs is known, although several necessary conditions and many suf- ficient conditions (see [6]) have been discovered. In fact, it is known that determining whether there are hamiltonian paths or cycles in arbitrary graphs is N P-complete. The interested reader is referred in particular to the surveys of Berge ([5], Chapter 10), Bondy and Murty ([10], Chapters 4 and 9), J. C. Bermond [6], Flandrin, Faudree and Ryj ́ a c ˇ stek [21] and R. Gould [27]. Hamiltonicity in special classes of graphs is a major area of graph theory and a lot of graph theorists have studied it. One special class of graphs whose hamiltonicity has been studied is that of Toeplitz graphs, introduced by van Dal et al. [13] in 1996. This study was continued by C. Heuberger [32] in 2002. The Toeplitz graphs investigated in [13] and [32] were all undirected. We intend to extend here this study to the directed case. A Toeplitz matrix, named after Otto Toeplitz, is a square matrix (n × n) which has constant values along all diagonals parallel to the main diagonal. Thus, Toeplitz matrices are defined by 2n − 1 numbers. Toeplitz matrices have uses in different areas in pure and applied mathematics, and also in computer science. For example, they are closely connected with Fourier series, they often appear when differential or inte- gral equations are discretized, they arise in physical data-processing applications, in viiviii the theories of orthogonal polynomials, stationary processes, and moment problems; see Heinig and Rost [31]. For other references on Toeplitz matrices see [26], [28] and A special case of a Toeplitz matrix is a circulant matrix, where each row is ro- tated one element to the right relative to the preceding row. Circulant matrices and their properties have been studied in [14] and [28]. In numerical analysis circulant matrices are important because they are diagonalized by a discrete Fourier trans- form, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. These matrices are also very useful in digital image processing. A directed or undirected graph whose adjacency matrix is circulant is called cir- culant. Circulant graphs and their properties such as connectivity, hamiltonicity, bipartiteness, planarity and colourability have been studied by several authors (see [8], [11], [15], [25], [35], [38], [41] and [24]). In particular, the conjecture of Boesch and Tindell [8], that all undirected connected circulant graphs are hamiltonian, was proved by Burkard and Sandholzer [11]. A directed or undirected Toeplitz graph is defined by a Toeplitz adjacency matrix. The properties of Toeplitz graphs; such as bipartiteness, planarity and colourability, have been studied in [18], [19], [20]. Hamiltonian properties of undirected Toeplitz graphs have been studied in [13] and [32]. For arbitrary digraphs the hamiltonian path and cycle problems are also very dif- ficult and both are N P-complete (see, e.g. the book [22] by Garey and Johnson). It is worthwhile mentioning that the hamiltonian cycle and path problems are N P- complete even for some special classes of digraphs. Garey, Johnson and Tarjan shows [23] that the problem remains N P-complete even for planar 3-regular digraphs. Some powerful necessary conditions, due to Gutin and Yeo [10], are considered for a digraphix to be hamiltonian. For information on hamiltonian and traceable digraphs, see e.g. the survey [2] and [3] by Bang-Jensen and Gutin, [9] by Bondy, [29] by Gutin and [39] by Volkmann. In this thesis, we investigate the hamiltonicity of directed Toeplitz graphs. The main purpose of this thesis is to offer sufficient conditions for the existence of hamil- tonian paths and cycles in directed Toeplitz graphs, which we will discuss in Chapters 3 and 4. The main diagonal of an (n × n) Toeplitz adjacency matrix will be labeled 0 and it contains only zeros. The n − 1 distinct diagonals above the main diago- nal will be labeled 1, 2, . . . , n − 1 and those under the main diagonal will also be labeled 1, 2, . . . , n − 1. Let s 1 , s 2 , . . . , s k be the upper diagonals containing ones and t 1 , t 2 , . . . , t l be the lower diagonals containing ones, such that 0 < s 1 < s 2 < · · · < s k < n and 0 < t 1 < t 2 < · · · < t l < n. Then, the corresponding di- rected Toeplitz graph will be denoted by T n s 1 , s 2 , . . . , s k ; t 1 , t 2 , . . . , t l . That is, T n s 1 , s 2 , . . . , s k ; t 1 , t 2 , . . . , t l is the graph with vertex set 1, 2, . . . , n, in which the edge (i, j), 1 ≤ i < j ≤ n, occurs if and only if j − i = s p or i − j = t q for some p and q (1 ≤ p ≤ k, 1 ≤ q ≤ l). In Chapter 1 we describe some basic ideas, terminology and results about graphs and digraphs. Further we discuss adjacency matrices, Toeplitz matrices, which we will encounter in the following chapters. In Chapter 2 we discuss hamiltonian graphs and add a brief historical note. We then discuss undirected Toeplitz graph, and finally mention some known results on hamiltonicity of undirected Toeplitz graphs found by van Dal et al. [13] and C. Heuberger [32].x Since all graphs in the main part of the thesis (Chapters 3 and 4) will be directed, we shall omit mentioning it in these chapters. We shall consider here just graphs without loops, because loops play no role in hamiltonicity investigations. Thus, un- less otherwise mentioned, in Chapters 3 and 4, by a graph we always mean a finite simple digraph. In Chapter 3, for k = l = 1 we obtain a characterization of cycles among directed Toeplitz graphs, and another result similar to Theorem 10 in [13]. Directed Toeplitz graphs with s 1 = 1 (or t 1 = 1) are obviously traceable. If we ask moreover that s 2 = 2, we see that the hamiltonicity of T n 1, 2; t 1 depends upon the parity of t 1 and n. Further in the same Chapter, we require s 3 = 3 and succeed to prove the hamiltonicity of T n 1, 2, 3; t 1 for all t 1 and n. In Chapter 4 we present a few results on Toeplitz graphs with s 1 = t 1 = 1 and s 2 = 3. They will often depend upon the parity of n. Chapter 5 contains some concluding remarks.
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