تاریخ کبیر میں امام بخاری کا رواۃ پر تنقید کے اسلوب کاجائزہ

Muḥadithīn have played a dynamic role in the preservation and compilation of Ḥadīth of the Prophet (SAW). Among them the contribution of Imam Bukhārī is unavoidable and remarkable in the field of Ḥadīth and its sciences. Since, he is known “Hafiz and Compiler” and considered as “Imam” and “authority” in the field of both higher and lower criticism. His book "Al-Tārīkh al Kabīr" (08 volumes) is considered as one of the ancient books in field of Ḥadīth and its sciences. This paper aims to describe the methodology of Imam Bukhārī in Al-Tārīkh al Kabīr. He described the biography of thirteen thousand seven hundred and seventy nine only (13779) in alphabetic order. Although his methodology was to mention names of the narrators alphabetically, but in respect with the Prophet (SAW), he gave place to eleven narrators among the companion whom names were started with the names of “Muhammad”. In the Methodology of Imam Bukhārī (RA), it is found that either the narrators are rejected or given authenticity, on the basis of not only his own examinations but he relied on the sayings of other Muḥadithīn as well. It is also pertinent to mention that he awarded degree to a little number of the narrators while left over most of the narrators undecided, because the main objective of the book was to portray the biographies of the narrators rather to make analysis of all the narrators.

Some Vertex-Degree-Based Topological Indices of Graphs

Many topological indices which are being used by researchers in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies to predict the physico-chemical properties of molecules, are based on vertex degrees of the corresponding molecular graphs. When a new topological index is introduced in chemical graph theory, one of the important questions that need to be answered is for which members of a certain class of n-vertex graphs this index assumes minimal and maximal values? On the other hand, there are many well known graph families and vertex-degree-based topological indices in the literature for which this question remains open. The main purpose of current study is to address this open question for some well known families of graphs. Firstly, the collection of all k-polygonal chains (for k = 3, 4, 5) with fixed number of k-polygons is considered and the extremal elements from this family are characterized with respect to several well known bond incident degree (BID) indices (BID indices form a subclass of the class of all vertex-degree-based topological indices). From the derived results, many already reported results are obtained as corollaries. Furthermore, the extremal 4-polygonal (polyomino) chains for some renowned vertex-degree-based topological indices (other than BID indices) are also determined. Next, the problem of characterizing the extremal cacti over the certain classes of cacti (tree-like polyphenylene systems, spiro hexagonal systems and general cacti) with some fixed parameters is addressed for various well known vertex-degree-based topological indices. Finally, some mathematical properties of the atom-bond connectivity index and augmented Zagreb index are explored