الشورى في الاسلام

Counseling is a vital pillar of Islamic Shariah by which teachings of Islam are interpreted to solve the modem day problems. It plays a crucial role in highlighting and resolving the collective issues of Muslim Ummah. The West has risen to this power and status because their thinktank body works actively. They have the current situation under their scrutiny and thus hold a guiding light for their nations. It is hightime for the Muslim Ummah to act proactively for the solution of multiple problems being faced by the Muslim Ummah. A think-tank body of the Muslim Ummah may not only unite the Muslim Ummah but it may pave the way for glorifying the Muslim Ummah. Such a think-tank shall give representation to the Muslim scholarly community.

Multilevel and Antipodal Labelings for Families of Graphs

A radio k-labeling c of a graph G is a mapping c : V (G) → Z+ ∪ {0}, such that d(x, y) + |c(x) − c(y)| ≥ k + 1 holds for every two distinct vertices x and y of G, where d(x, y) is the distance between any two vertices x and y of G. The span of a radio k-labeling c is denoted by sp(c) and defined as max{|c(x) − c(y)| : x, y ∈ V (G)}. The radio labeling is a radio klabeling when k = diam(G). In other words, a radio labeling is a one-to-one function c : V (G) → Z+ ∪ {0}, such that |c(x) − c(y)| ≥ diam(G) + 1 − d(x, y) for any pair of vertices x, y in G. The radio number of G denoted by rn(G), is the lowest span taken over all radio labelings of the graph. When k = diam(G) − 1, a radio klabeling is called a radio antipodal labeling. An antipodal labeling for a graph G is a function c : V (G) → {0, 1, 2, ...}, so that d(x, y) + |c(x) − c(y)| ≥ diam(G) for all x, y ∈ G. The radio antipodal number for G denoted by an(G), is the minimum span of an antipodal labeling admitted by G. In this thesis, we investigate the exact value of the radio number and radio antipodal number for different family of graphs. Further more, we also determine the lower bound of the radio number for some cases.