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مولاناابوالوفاء افغانی

مولانا ابو الوفاء افغانی
معارف کے قارئین مولانا ابوالوفاء افغانی کے نام سے اور کام سے بخوبی واقف ہیں ان کی پیدائش افغانستان میں ہوئی، لیکن تعلیمی مراحل ہندوستان میں طے ہوئے اور یہاں کے باکمال اساتذہ سے استفادہ کیا، زندگی کا بڑا حصہ حیدرآباد میں گزرا، عرصہ تک مدرسہ نظامیہ میں مدرس رہے، فقہ حنفی سے خاص مناسبت تھی اور امام ابوحنیفہ ان کے تلامذہ اور ائمہ احناف سے بے حد عقیدت تھی، لجنۃ احیاء المعارف العثمانیہ ان کی اس عقیدت کی گواہ ہے، انھوں نے قاضی ابو یوسف امام محمد اور شمس لائمہ سرخسی کی کتابیں ڈھونڈ کر جمع کیں، بڑی دیدہ ریزی کے ساتھ مختلف نسخوں کا مقابلہ کیا، جان کا ہی کے ساتھ ان کی تصحیح کی، حواشی لکھے، انڈکس بنائے اور حسن و خوبی کے ساتھ طباعت کا انتظام کیا، وہ پرانے مدرسوں کے پڑھے ہوئے تھے، لیکن نئے محققین بھی ان کا لوہا مانتے تھے، افسوس ہے کہ گزشتہ ماہ علم و تحقیق کی یہ شمع خاموش ہوگئی، اﷲ تعالیٰ ان کو اپنی رحمتوں سے سرفراز فرمائے اور ان کے بیش بہاکاموں کے جاری رکھنے کا انتظام فرمائے۔
(عبد السلام قدوائی ندوی، ستمبر ۱۹۷۵ء)

 

The Impact of Prophetic Hijrah on the Societies of Makkah and Madinah

The aim of Hijrah is philosophically viewed as a way to spread Islam wisely. This was by Allah’s guidance and directive amidst the very bitter atmosphere in Makkah at that time against the early followers of the Islamic religion brought by the Rasulullah (pbuh). The prophetic Hijrah observed from the way it impacted Makkah and Madinah’s societies can be construed as a very important event that provided us with a variety of messages. It also acts as a reflection of our contributions to da’wah and the roles we can assume as good Muslims, by looking at what the Rasulullah (pbuh) and his companions had themselves sacrificed in the Hijrah. In short, Al-Hijra is the time when the Prophet Muhammad (pbuh) and his followers moved from Makkah to Madinah, where they set up the first Islamic state. Islam needed to expand and spread in the world, so the migration resulted in the expansion and preservation of Islam and Muslims. The Rasulullah (pbuh) migrated to Madinah when his enemies in Makkah mistreated him and his followers. It is a fact that the early Muslims in Makkah were greatly troubled by the unbelievers of Quraisy, the tribe of the Prophet (pbuh). The Rasulullah’s (pbuh) popularity in his da'wah efforts to invite his people to Islam was seen as threatening by the people in power in Makkah. The context of Hijrah was seen as urgent and timely as the unbelievers in Makkah had escalated the persecution against Muhammad (pbuh) and his followers. This persecution and a directive from Allah were the main reasons for the migration.

Hamiltonian Properties of Generalized Halin Graphs

A Halin graph is a graph H = T ∪ C, where T is a tree with no vertex of degree two, and C is a cycle connecting the end-vertices of T in the cyclic order determined by a plane embedding of T . Halin graphs were introduced by R. Halin [16] as a class of minimally 3-connected planar graphs. They also possess interesting Hamiltonian properties. They are 1-Hamiltonian, i.e., they are Hamiltonian and remain so after the removal of any single vertex, as Bondy showed (see [23]). Moreover, Barefoot proved that they are Hamiltonian connected, i.e., they admit a Hamiltonian path be- tween every pair of vertices [1]. Bondy and Lov ́asz [6] and, independently, Skowronska [33] proved that Halin graphs on n vertices are almost pancyclic, more precisely they contain cycles of all lengths l (3 ≤ l ≤ n) except possibly for a single even length. Also, they showed that Halin graphs on n vertices whose vertices of degree 3 are all on the outer cycle C are pancyclic, i.e., they must contain cycles of all lengths from 3 to n. In this thesis, we define classes of generalized Halin graphs, called k-Halin graphs, and investigate their Hamiltonian properties. In chapter 4, we define k-Halin graph in the following way. A 2-connected planar graph G without vertices of degree 2, possessing a cycle C such that (i) all vertices of C have degree 3 in G, and (ii) G − C is connected and has at most k cycles is called a k-Halin graph. A 0-Halin graph, thus, is a usual Halin graph. Moreover, the class of k-Halin graphs is contained in the class of (k + 1)-Halin graphs (k ≥ 0). We shall see that, the Hamiltonicity of k-Halin graphs steadily decreases as k increases. Indeed, a 1-Halin graph is still Hamiltonian, but not Hamiltonian con- nected, a 2-Halin graph is not necessarily Hamiltonian but still traceable, while a 3-Halin graph is not even necessarily traceable. The property of being 1-Hamiltonian, Hamiltonian connected or almost pancyclic is not preserved, even by 1-Halin graphs. However, Bondy and Lov ́asz’ result about the pancyclicity of Halin graphs with no inner vertex of degree 3 remains true even for 3-Halin graphs. The property of being Hamiltonian persists, however, for large values of k in cubic 3-connected k-Halin graphs. In chapter 5, it will be shown that every cubic 3- connected 14-Halin graph is Hamiltonian. A variant of the famous example of Tutte [37] from 1946 which first demonstrated that cubic 3-connected planar graphs may not be Hamiltonian, is a 21-Halin graphs. The cubic 3-connected planar non-Hamiltonian graph of Lederberg [21], Bos ́ak [7] and Barnette, which has smallest order, is 53-Halin. The sharpness of our result is proved by showing that there exist non-Hamiltonian cubic 3-connected 15-Halin graphs.
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