### $$λ$$-fractional schwarzian derivative and $$λ$$-fractional mobius transformation

JPRM-Vol. 1 (2010), Issue 1, pp. 56 – 61 Open Access Full-Text PDF
Y. Polatoglu
Abstract: We denote by A the class of all analytic functions in the open unit disk $$\mathbb{D} = {z | |z| < 1}$$ which satisfy the conditions $$f(0) = 0$$, $$f'(0) = 1$$. In this paper we define a new concept of $$λ$$− fractional Schwarzian derivative and $$λ$$− fractional Mobius transformation for the class A. We also formulate the criterion for a function to be univalent using the fractional Schwarzian.

### Some exact solutions for the flow of a Newtonian fluid with heat transfer via prescribed vorticity

JPRM-Vol. 1 (2010), Issue 1, pp. 38 – 55 Open Access Full-Text PDF
M. Jamil, N. A. Khan, A. Mahmood, G. Murtaza, Q. Din
Abstract: Two-dimensional , steady, laminar equations of motion of an incompressible fluid with variable viscosity and heat transfer equations are considered. The problem investigated is the flow for which the vorticity distribution is proportional to the stream function perturbed by a sinusoidal stream. Employing transformation variable, the governing Navier-Stokes Equations are transformed into the ordinary differential equations and exact solutions are obtained. Finally, the influence of different parameters of interest on the velocity, temperature and pressure profiles are plotted and discussed.

### Some distributional properties of the concomitants of record statistics for bivariate pseudo–exponential distribution and characterization

JPRM-Vol. 1 (2010), Issue 1, pp. 32 – 37 Open Access Full-Text PDF
Abstract: A new class of distributions known as Bivariate Pseudo–Exponential distribution has been defined. The distribution of r–th concomitant and joint distribution of r–th and s–th concomitant of record statistics of the resulting distribution have been derived. Expression for single and product moments has also been obtained for the resulting distributions. A characterization of the k-th concomitant of record statistics for the Pseudoexponential distribution by the conditional expectation is presented.

### Graphs with same diameter and metric dimension

JPRM-Vol. 1 (2010), Issue 1, pp. 22 – 31 Open Access Full-Text PDF
Imrana Kousar, Ioan Tomescu, Syed Muhammad Husnine
Abstract: The cardinality of a metric basis of a connected graph $$G$$ is called its metric dimension, denoted by $$dim(G)$$ and the maximum value of distance between vertices of $$G$$ is called its diameter. In this paper, the graphs $$G$$ with diameter 2 are characterized when $$dim(G) = 2.$$

### Forcing edge detour number of an edge detour graph

JPRM-Vol. 1 (2010), Issue 1, pp. 13 – 21 Open Access Full-Text PDF
A. P. Santhakumaran, S. Athisayanathan
Abstract: For two vertices $$u$$ and $$v$$ in a graph $$G = (V, E)$$, the detour distance $$D(u, v)$$ is the length of a longest $$u–v$$ path in $$G$$. A $$u–v$$ path of length $$D(u, v)$$ is called a $$u–v$$ detour. A set $$S ⊆ V$$ is called an edge detour set if every edge in $$G$$ lies on a detour joining a pair of vertices of $$S$$. The edge detour number $$dn_1(G)$$ of G is the minimum order of its edge detour sets and any edge detour set of order $$dn_1(G)$$ is an edge detour basis of $$G$$. A connected graph $$G$$ is called an edge detour graph if it has an edge detour set. A subset $$T$$ of an edge detour basis $$S$$ of an edge detour graph $$G$$ is called a forcing subset for $$S$$ if $$S$$ is the unique edge detour basis containing $$T$$. A forcing subset for $$S$$ of minimum cardinality is a minimum forcing subset of $$S$$. The forcing edge detour number $$fdn_1(S)$$ of $$S$$, is the minimum cardinality of a forcing subset for $$S$$. The forcing edge detour number $$fdn_1(G)$$ of $$G$$, is $$min{fdn_1(S)}$$, where the minimum is taken over all edge detour bases $$S$$ in $$G$$. The general properties satisfied by these forcing subsets are discussed and the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters $$dn_1(G)$$ and $$fdn_1(G)$$ satisfy the relation $$0 ≤ fdn_1(G) ≤ dn_1(G)$$ and it is proved that for each pair $$a$$, $$b$$ of integers with $$0 ≤ a ≤ b$$ and $$b ≥ 2$$, there is an edge detour graph $$G$$ with $$fdn_1(G) = a$$ and $$dn_1(G) = b$$.
Abstract: A decomposition of a graph G is a collection Ψ of edge-disjoint subgraphs $$H_1,H_2, . . . , H_n$$ of $$G$$ such that every edge of $$G$$ belongs to exactly one $$H_i$$. If each $$H_i$$ is a path in $$G$$, then $$Ψ$$ is called a path partition or path cover or path decomposition of $$G$$. A divisor path decomposition of a $$(p, q)$$ graph $$G$$ is a path cover $$Ψ$$ of $$G$$ such that the length of all the paths in $$Ψ$$ divides $$q$$. The minimum cardinality of a divisor path decomposition of $$G$$ is called the divisor path decomposition number of $$G$$ and is denoted by $$π_D(G)$$. In this paper, we initiate a study of the parameter $$π_D$$ and determine the value of $$π_D$$ for some standard graphs. Further, we obtain some bounds for $$π_D$$ and characterize graphs attaining the bounds.