Let G = (V (G);E(G)) be a connected graph. The distance between two vertices u; v 2 V (G) is the length of shortest path between them and is denoted by d(u; v). A vertex x is said to resolve a pair of vertices u; v 2 V (G) if d(u; x) 6= d(v; x). For an ordered subset, B = fb1; b2; : : : ; bng of vertices of G, the n-tuple r(vjB) = (d(v; b1); d(v; b2); : : : ; d(v; bn)) is called representation of vertex v with respect to B or vector of metric coordinates of v with respect to B. The set B is called a resolving set of G if r(ujB) 6= r(vjB) for every pair of vertices u; v 2 V (G), i.e., the representation of each vertex with respect to B is unique. The resolving set with minimum cardinality is called metric basis of G. This minimum cardinality is called metric dimension and is denoted by _(G). Notice that the i-th coordinate in r(vjB) is 0 if and only if v = bi. Thus in order to show that B is a resolving set of G, it su_ces to verify that r(ujB) 6= r(vjB) for every pair of distinct vertices u; v 2 V (G) n B. Let G be a graph of order at least 2. Two vertices x; y 2 V (G) are said to doubly resolve the vertices u; v of G if d(u; x) ? d(u; y) 6= d(v; x) ? d(v; y): A subset D _ V (G) is called a doubly resolving set of G if every two distinct vertices of G are doubly resolved by some two vertices in D, i.e., all coordinates of the vector r(ujD)?r(vjD) can not be same for every pair of distinct vertices u; v 2 V (G). The minimal doubly resolving set problem is to _nd a doubly resolving set of G with the minimum cardinality. The cardinality of minimal doubly resolving set of G is denoted by(G). We have _(G) _(G) always. Therefore these sets can contribute in finding upper bounds on the metric dimension of graphs. In this thesis, we have investigated the minimal doubly resolving set problem for necklace graph, circulant graph, antiprism graph and M obius ladders. Also, in last part of thesis, the metric dimension problem has been investigated for kayak paddle graph and cycles with chord.
This article probes into poetical citation in the historical letter of Ibn-e Zaydun, a renowned Andalusion poet of 11th century A.D. Ibn-e Zaydun was imprisoned by king of Córdoba, Ibn-e- Jahoor. While in prison, Ibn-e- Zaydun wrote Ibn- e- Jahoor a letter lamenting that he has been thrown into prison for no reason and appealed for mercy and leniency towards him. The depth of thoughts reflected in the poetic text of Ibn- e- Zaydun`s letter testifying his command over poetry. The poet who is quoted in the letter of Ibn- e- Zaydun is known as Al- Mutanabi. The article examines the parts of the Ibn- e- Zaydun`s letter citing the poetry of Al- Mutanabi in order to make it effective in achieving the objectives of the study.
Language use in academic discourse i.e. in classroom sessions and conference presentations is a controversial issue among linguists and academicians as they carry contrasting perspectives regarding use of language in academia. Some linguists believe that code-switching is an essential part of academic discourse while some other suggests that mutual intelligibility may not be possible if the learners switch their language during communication. The variation of the viewpoints creates a niche to explore the use of language in Pakistani classroom sessions and conference presentations. This study explores various purposes of code-switching in academic discourse including elucidation, giving instruction, translation, change/introduce the topic, asking question and building argument. Pakistan is a multi-lingual country and it has rich linguistic diversity where people use provincial and regional languages and medium of instruction in classrooms is a serious concern for academicians. Moreover, the study portrays present situation and describes future implications from students’ perspective.The second main objective of the study is to explore the contextual relevance, quantity of information and perspicuousness in academic discourse (classroom session and conference presentation). To conduct this study thirty classroom sessions and forty conference presentations were recorded and transcribed. Furthermore, this research highlights the issues related to contextual use of language, quantity of information and clarity of expression in academic discourse by giving examples of transcribed data. The findings reveal the purposes of code-switching and contextual uses of language in academic discourse with specific reference to pragmatic ideology. In the light of the findings of the current study I propose Relative Relevance Model of Communication which has potential to explore relevance and relative relevance in classroom sessions and conference presentations especially. Moreover, this model may also be employed on other genres in order to address pragmatic manifestations with special focus on relevance.