## Issue 4Volume 4 (2016)

ISSUE 4 Special Issue
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 Article Type : Research Article Title : Certain Combinatorial Properties of Twin Triplets Related to Tchebychev Polynomials Country : India Authors : R.Rangarajan, P.Shashikala and C.K.Honnegowda

Abstract: In the present paper, Tchebychev polynomials $U_{n}(x)$, $V_{n}(x)=U_{n}(x)-\,U_{n-1}(x)$ and $W_{n}(x)=U_{n}(x)+\,U_{n-1}(x)$ are extended to two variables. Twin triplets of numbers $(y_{n},d_{n},s_{n})$ and $(Y_{n},D_{n},S_{n})$ are defined and their certain combinatorial properties are described.

Keywords: Combinatorial Identities, Continued fractions and Functions of hypergeometric type in one and severable variables.

C.K.Honnegowda
Department of Studies in Mathematics, University of Mysore, Mysore, India.
E-mail: honnegowdack@gmail.com

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 Article Type : Research Article Title : Odd Sum Labeling of Tree Related Graphs Country : India Authors : R.Gopi

Abstract: An injective function $f:V(G)\to \{ 0,1,2,...,q\}$ is an odd sum labeling if the induced edge labeling $f^{*}$ defined by $f^{*} (uv)=f(u)+f(v)$ for all $uv\in E(G)$ is a bijective and $f^{*} (E(G))=\{ 1,3,5,...,2q-1\}$. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we investigate odd sum labeling of some more graphs.

Keywords: Odd sum Labeling, Odd Sum Graph, Twig graph, H-graph.

R. Gopi
PG and Research Department of Mathematics, Srimad Andavan Arts and Science College(Autonomous), Tiruchirappalli, Tamil Nadu, India.
E-mail: drrgmaths@gmail.com

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 Article Type : Research Article Title : M/G/1 Queue with Multiple Optional Services and Deterministic Repair Times Country : India Authors : S.Vanitha

Abstract: We analyze the steady state behavior of an $M/G/1$ queue with Poisson arrivals subject to multiple optional services and random system breakdowns. Arriving customer has to undergo first essential service and there are $j$ optional services, where $j=1,2,\ldots, n$. As soon as the essential service of a customer is complete, then with probability $r_{j},j=1\ldots n$, he may opt for any one of the $j$ optional services, in which case his any one of the $j$ services will immediately commence or else with probability $1-\sum\limits_{j=1}^{n}r_{j}$, he may opt to leave the system, in which case another customer at the head of the queue is taken up for his essential service.The service times follow arbitrary (general) service distributions. The system is prone to random breakdowns and just after a breakdown the server undergoes repair of a fixed duration. We obtain time dependent as well as steady state probability generating functions for the number in the system. For steady state we obtain explicitly the mean number and the mean waiting time for the system and for the queue. Results for some special cases of interest are derived.

Keywords: Poisson arrivals, general service times, probability generating functions, random breakdowns, idle state, steady state, deterministic repairs, supplementary variable technique.

S.Vanitha
SSN College of Engineering, Old Mahabalipuram Road, Kalavakkam, Chennai, India.
E-mail: vanithas@ssn.edu.in

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 Article Type : Research Article Title : Magnetohydrodynamic Flow of Viscous Incompressible Fluid Surrounded by a Porous Media Between Two Non-Coincident Rotating Disks Country : India Authors : Divya Saxena and V.K.Sharma

Abstract: Analytical investigations of the Navier-Stokes Equations examine the magnetohydrodynamic flow of a viscous incompressible fluid between two non-coincident disks rotating with the same angular velocity $\Omega$. The governing equation of motions are coupled non linear partial differential equations together with the boundary conditions, are reformed into the linear ordinary differential equations that has been studied with the assumptions. The results have been obtained for the velocities, shearing stress and torque for several values of rotational parameter, magnetic parameter and porosity parameter. The effects of the several parameters on the flow field are presented and explained graphically.

Keywords: Magnetohydrodynamic Flow, Non-Coincident Disk, Porous Medium, Magnetic Parameter, Rotation Parameter.

Divya Saxena
Department of Mathematics, Jagannath University, Jaipur, India.
E-mail: divyasaxena0516@gmail.com

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 Article Type : Research Article Title : Common Fixed Point Theorems in Multiplicative Metric Spaces satisfying E.A. property and (CLR) property for Rational Contractive Map Country : India Authors : Manoj Kumar, Kamal Kumar, Nisha Sharma, Rajeev Jha and Arti Mishra

Abstract: In this paper we prove common fixed point theorems for weakly compatible mappings along with E.A property and common limit range properties using Rational contraction satisfying implicit functions in multiplicative metric space.

Keywords: E.A. property, common limit range properties, implicit function, common fixed point, multiplicative metric space.

Manoj Kumar
Department of Mathematics, Lovely Professional University, Punjab, India.
E-mail: manojantil18@gmail.com

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 Article Type : Research Article Title : Partitioned q-k-Ep Matrices Country : India Authors : K.Gunasekaran and K.Gnanabala

Abstract: Necessary and sufficient conditions are determined for a schur complement in a q-k-EP matrix to be q-k-EP. Further it is shown that in q-k-EPr matrix, every principal sub matrix of rank r is q-k-EPr. Necessary and sufficient conditions for products of q-k-EPr partitioned matrices to be q-k-EPr is given.

Keywords: q-k-EP matrices, Schur complements.

K.Gnanabala
Ramanujan Research centre, PG and Research Department of Mathematics, Government Arts College(Autonomous), Kumbakonam, India.
E-mail: gbalamaths@gmail.com

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 Article Type : Research Article Title : Minkowski inverse for the Range symmetric block matrix with two identical sub-blocks over skew fields in Minkowski space $\mathscr M$ Country : India Authors : D.Krishnaswamy and S.Anusuya

Abstract: Let $K$ be a skew field and $K^{n\times n}$ be the set of all $n \times n$ matrices over $K$. The purpose of this paper is to give some necessary and sufficient conditions for the existence and the representations of the minkowski inverse of the block matrix $\begin{pmatrix}A&B\\C&D \end{pmatrix}$ under some conditions.

Keywords: Skew fields, Block matrix, Minkowski inverse.

S.Anusuya
Department of Mathematics, Annamalai University, Annamalai Nagar. Tamilnadu, India.
E-mail:

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 Article Type : Research Article Title : Common Fixed Point Theorem in Rational Inequality and Their Application Country : India Authors : Madhu Shrivastava, K.Qureshi and A.D.Singh

Abstract: In this paper we prove some sufficient condition for the existence and uniqueness of fixed point and common fixed point in rational inequality on complete metric spaces. As application, Some existence and uniqueness results of solution and common solution for some functional equations and system of functional equations in Dynamic programming are given by using the fixed point and common fixed point theorems.

Keywords: Common fixed point, complete metric space, common solution, functional equation, system of functional equations, dynamic programming.

Department of Mathematics, TIT Group of Institution, Bhopal, India.

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 Article Type : Research Article Title : Extended Diffie-Hellmann Protocol Based on Lucas Actions Country : India Authors : P.Anuradha Kameswari, S.Uma Devi and T.Surendra

Abstract: Discrete log problem (DLP) is a special case of semigroup action problem (SAP) that leads to generalized DLP. The cryptographic protocols like Diffie-Helman/ Elgamal based on DLP may be extended using semigroup action problem (SAP). The effect of square root attacks like Pollard Rho, Shanks baby step - giant step on these extensions with group actions, led to the study of a square root attacks on the generalized DLP, in the cases when, the semigroup has a large sub group and with Matrix action on abelian groups. We propose to extend the study of square root attack, Shank's Baby-step Giant-step on the generalized DLP considering the special case of LUCAS actions on abelian groups.

Keywords: Lucas group actions, Lucas Semigroup actions, Generalized discrete log problems, Extended Diffie-Hellmann protocol.

Department of Mathematics, Andhra University, Visakhapatnam, India.

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 Article Type : Research Article Title : Performance Analysis of Two Echelon Perishable Inventory System with Joint Ordering Policy Country : India Authors : R.Karthikeyan and K.Krishnan

Abstract: Perishable inventory system has received increasing attentions in the past years. An optimal ordering policy for deteriorating inventory system has become more and more important. This paper develops a model to determine the optimal order quantity that minimizes the total expected cost for perishable items. In addition, the solution methodologies as well as an analysis of results are presented. In this study, we create a scenario where two types of products are considered with two locations (Distributor and Retailer) in such a way that there exists a joint ordering policy. A continuous review perishable inventory model with independent Poisson demand for items at retailer and a direct Poisson demand at distributor is assumed. The items are supplied to the retailers from the distribution center (DC) administrated with exponential lead time. The joint probability disruption of the inventory levels of two products at retailer and the supplier are obtained in the steady state case. Various system performance measures are derived and the long run total expected inventory cost rate is calculated. Several instances of numerical examples, which provide insight into the behavior of the system, are presented.

Keywords: Perishable inventory, Two-echelon, Joint ordering policy, Optimization.

K.Krishnan
PG \& Research Department of Mathematics, Cardamom Planters' Association College, Bodinayakanur, Tamil Nadu, India.
E-mail: drkkmaths@gmail.com

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 Article Type : Research Article Title : The Split (Nonsplit) Nomatic Number of a Graph Country : India Authors : B.Chaluvaraju and C.Nandeeshkumar

Abstract: For a given connected graph $G = (V, E)$, a set $S \subseteq V(G)$ is a neighborhood set of $G$, if $G = \bigcup\limits_{v\in S}\langle N[v]\rangle$, where $\langle N[v]\rangle$ is the sub graph of $G$ induced by $v$ and all vertices adjacent to $v$. A neighborhood set $S$ is a split (nonsplit) neighborhood set if $\langle V(G)- S \rangle$ is connected (disconnected). The maximum number of a partition of $V(G)$, all of whose are split (nonsplit) neighborhood sets, is the split(nonsplit) nomatic number $N_{s}(G)(N_{ns}(G))$. Our purpose in this paper is to initiate the study of split(nonsplit) nomatic number of a graph. We first study basic properties and bounds for $N_{s}(G)(N_{ns}(G))$. In addition, we determine the $N_{s}(G)(N_{ns}(G))$ of some classes of graphs.

Keywords: Graph, domatic, nomatic, split nomatic, nonsplit nomatic.

B.Chaluvaraju
Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore, India.
E-mail: bchaluvaraju@gmail.com

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 Article Type : Research Article Title : Edge-Wiener Indices of $n$-circumscribed Peri-condensed Benzenoid Graphs Country : India Authors : Jasintha Quadras, K.Arputha Christy and A.Nelson
Abstract: The cut method demonstrates its usefulness especially for the topological indices that are based on the distances in the molecular graphs without actually calculating the distances between pairs of vertices. The Wiener index is equal to the sum of distances between all pairs of vertices of the connected graph $G$, whereas the Edge-Wiener index is the sum of distances between all pairs of edges of the connected graph $G$. In this paper we calculate the Edge-Wiener indices of Circum-polyacenes, Circum-pyrenes and Circum-trizenes.