## Issue 4Volume 5 (2017)

ISSUE 4− A ISSUE 4− B ISSUE 4− C ISSUE 4 − D ISSUE 4 − E ISSUE 4 − F
 Authors : B.C. Dhage, N.S. Jadhav, J.N. Salunke and A.Y. Shete Title : Dhage Iteration Method for Nonlinear First Order Hybrid Functional Differential Equations of Second Type Linear Perturbations Volume, Issue, Year : 5(4−E)(2017) Pages : 605−614

Abstract: In this paper we prove the existence and approximation result for a first order nonlinear initial value problem of hybrid functional differential equations via construction of an algorithm. The main results rely on the Dhage iteration method embodied in a recent hybrid fixed point principle of Dhage (2014). An example is also furnished to illustrate the hypotheses and the abstract result of this paper.

Keywords: Hybrid functional differential equation, Dhage iteration method, Existence and Approximation theorem.

 Authors : B.C. Dhage, A.D. Kadam, J.N. Salunke and A.Y. Shete Title : On Some Fixed Point Theorems for Generalized Contractive Mappings in an Euclidean Space $\mathbb{R}^n$ Volume, Issue, Year : 5(4−E)(2017) Pages : 615−618

Abstract: In this paper a couple of fixed point theorems for contraction and Kannan mappings are proved in an Euclidean space $\mathbb{R}^n$ via calculus method and using the max/mini principle. It is shown that though our approach to Banach and Kannan mappings is different from the constructive one, we are not far away from the usual method. Actually one can take an arbitrary point $x_0\in \mathbb{R}^n$ and can define a sequence $\{x_n\}$ of iterates of the mapping under consideration. Then it is shown that the sequence converges to a fixed point geometrically.

Keywords: Contraction map, Fixed point theorem, max/mini principle.

 Authors : K.P.R. Sastry, K.K.M. Sarma, P. Krishna Kumari and Sunitha Choudari Title : Fixed Point Theorems for $\psi$-contractions in S-metric Spaces and KKS-metric Spaces Volume, Issue, Year : 5(4−E)(2017) Pages : 619−632

Abstract: In this paper we introduce the notion of $\psi$-contractions for self maps on S-metric spaces and establish a fixed point theorem for such maps. We also introduce a new subclass of S-metric spaces called KKS metric spaces and prove a fixed point theorem for a $\psi$-contraction on a KKS metric space. An open problem is also given at the end of the paper.

Keywords: S-metric space, KKS metric space, $\psi$-contraction.

 Authors : M.V. Subba Rao, M.N.R. Chowdary and J. Vijayasekhar Title : Two-person Zero-sum Game Problem Solution: Integer Simplex Method Volume, Issue, Year : 5(4−E)(2017) Pages : 633−635

Abstract: In this paper, we have calculated the optimal strategies of two-person zero-sum game problems which does not have a saddle point using Integer simplex method (Gomory's cutting plane method).

Keywords: Linear programming problem (LPP), Game problem, optimal strategies, Integer simplex method (Gomory's cutting plane method).

 Authors : Surajit Dutta Title : A New Construction of Apollonius Circle and a New Proof of Secant-Tangent Theorem Volume, Issue, Year : 5(4−E)(2017) Pages : 637−639

Abstract: Here in this paper we give an easy and elegant construction of Apollonius circle by using a simple property of isosceles triangles. We also give a simple proof of the Secant-Tangent theorem by using the same property of isosceles triangles.

Keywords: Apollonius Circle, Secant-Tangent Theorem.

 Authors : G.C. Basavaraju, M. Vishukumar and P. Raghunath Title : Metro Domination of Square Cycle Volume, Issue, Year : 5(4−E)(2017) Pages : 641−645

Abstract: Let $G=(V,E)$ be a graph.A set $S\subseteq V$ is called resolving set if for every $u,v\in V$ there exist $w\in V$, such that $d(u,w)\neq d(v,w)$. The resolving set with minimum cardinality is called metric basis and its cardinality is called metric dimention and it is denoted by $\beta (G)$. A set $D\subseteq V$ is called dominating set if every vertex not in $D$ is adjacent to at least one vertex in $D$. The dominating set with minimum cardinality is called domination number of $G$ and it is denoted by $\gamma (G)$. A set which is both resolving set as well as dominating set is called metro dominating set. The minimum cardinality of a metro dominating set is called metro domination number of $G$ and it is denoted by $\gamma_ \beta(G)$. In this paper we determine metro domination number of square cycle.

Keywords: Power graph, metric dimension, landmark, distance matrix.

 Authors : Amar Nath and Manohar Singh Chahar Title : Study and Analysis of Almost Para Sasakian Type-Riemannian Manifold Volume, Issue, Year : 5(4−E)(2017) Pages : 647−654

Abstract: In this paper, author studied and analyses Almost Para Sasakian Type-Riemannian manifold. The first section of this paper is introductory in nature, which deals with basic definition and literature review with previous known defined results. Second section deals with Almost Para Sasakian Type-Riemannian manifold (APST-Riemannian manifold) and its various applications and the third section is devoted for Para-K-contact type Riemannian manifold (PKCT-Riemannian manifold). As the outcomes of this work further detail theorems are suggested for future work.

Keywords: APST-Riemannian manifold, PQST manifold, PKCT-Riemannian manifold, tensor analysis, differential geometry.

 Authors : N.B. Rathod and K.K. Kanani Title : $k$-cordial Labeling of Triangular Belt, Alternate Triangular Belt, Braid Graph and $Z$-$P_n$ Volume, Issue, Year : 5(4−E)(2017) Pages : 655−662

Abstract: In this research paper, we find $k$-cordial labeling of some special graphs. We prove that Triangular Belt $TB(n)$$(\downarrow^n), Alternate Triangular Belt ATB(n)$$(\downarrow\uparrow\downarrow\uparrow...)$ and Braid Graph $B(n)$ are $k$-cordial. Moreover, the graph $Z$-$P_{n}$ is $k$-cordial for all odd $k$.

Keywords: Abelian Group, $k$-cordial Labeling, Triangular Belt, Alternate Triangular Belt, Braid graph, $Z$-$P_{n}$.

 Authors : V.R.Kulli Title : Revan Indices of Oxide and Honeycomb Networks Volume, Issue, Year : 5(4−E)(2017) Pages : 663−667

Abstract: There are many types of topological indices. Among degree based topological indices, Zagreb indices, Banhatti indices, Gourava indices are studied well. In this paper, we introduce the Revan indices and compute exact formulas for oxide and honeycomb networks.

Keywords: Revan indices, molecular graph, oxide network, honeycomb network.

 Authors : Hongyan Guan and Yan Hao Title : Products of Toeplitz and Hankel Operators on the Harmonic Bergman Space Volume, Issue, Year : 5(4−E)(2017) Pages : 669−673

Abstract: In this paper, we discuss the product problems of Toeplitz operators, small Hankel and big Hankel operators with quasihomogeneous symbols on the harmonic Bergman space of the unit disk.

Keywords: Toeplitz operator, small Hankel operator, big Hankel operator, quasihomogeneous symbols, harmonic Bergman space.

 Authors : D. Rajan and D. Vijayabalan Title : Some Characterization of Fuzzy Hazard Rate Order and Fuzzy Proportional Hazard Rate Order Volume, Issue, Year : 5(4−E)(2017) Pages : 675−684

Abstract: In this paper, we have recalled some of the known stochastic orders and the shifted version of them, so discussed their relations. Also, we obtained some applications of proportional hazard rate ordering.

Keywords: Fuzzy random variables, Fuzzy Hazard rate order, Shifted fuzzy Hazard rate order.

 Authors : M. Rameshpandi, I. Rajasekaran and O. Nethaji Title : Strong Form of Some Nano Open Sets Volume, Issue, Year : 5(4−E)(2017) Pages : 685−691

Abstract: The purpose of this paper, nano $\mathcal{I_R}$-set, nano $\mathcal{R_S}$-set, nano pre-regular, nano $\beta$-regular and nano weak $\mathcal{O_N}$-set are introduced and investigated on the line of research.

Keywords: Nano $\mathcal{I_R}$-set, nano $\mathcal{R_S}$-set, nano pre-regular, nano $\beta$-regular and nano weak $\mathcal{O_N}$-set.

 Authors : I. Rajasekaran, R. Bhavani and O. Nethaji Title : On Nano $\wedge_{rg}$-closed Sets Volume, Issue, Year : 5(4−E)(2017) Pages : 693−698

Abstract: In this paper, we introduce nano $\wedge_{rg}$-closed sets in nano topological spaces. Some properties of nano $\wedge_{rg}$-closed sets and nano $\wedge_{rg}$-open sets are weaker forms of nano closed sets and nano open sets.

Keywords: Nano $\wedge_r$-set, nano $\lambda_r$-closed set and nano $\wedge_{rg}$-closed set.

 Authors : K.B.Murthy and B.V.Dhananjayamurthy Title : On Domatic Energy of a Graph Volume, Issue, Year : 5(4−E)(2017) Pages : 699−709

Abstract: In this paper, Motivated by the partition energy and the minimum covering energy of a graph, we introduce domatic energy $E_d\left(G\right)$, of a graph $G$.

Keywords: Domatic Partition, domatic matrix, domatic eigenvalues, domatic energy of a graph.

 Authors : K.Anitha, S.Maragathasundari and M.Bala Title : Queuing System of Bulk Arrival Model With Optional Services in Third Stage and Two Different Vacation Policies Volume, Issue, Year : 5(4−E)(2017) Pages : 711−721

Abstract: This paper presents an introduction into the most important queueing theory concepts used for modeling computer performance. In this paper, customers arrive in batches to the queuing system. Arrival of customers follows a poisson process. The service to the customers is given in three stages in which the first two stages are compulsory and the third stage is optional. Server can avail vacation of short or long period after the completion of service. Moreover, the server may breakdown at random which leads to a repair process immediately. By means of supplementary variable technique and generating function approach the probability generating function of the cloud computing user queue size is determined. The other performance measures of the model are derived by means of Little's law. A trade-off between the Quality of service and queue length is observed.

Keywords: Stages of service, optional stage of service, service interruption,short vacation and long vacation.

 Authors : V. J. Kaneria and H. P. Chudasama Title : Absolute Mean Graceful Labeling in Various Graphs Volume, Issue, Year : 5(4−E)(2017) Pages : 723−726

Abstract: In the pesent study of paper, we defined absolute mean graceful labeling for various graphs. We proved all path graphs $P_{n}$, cycle $C_{n}$, complete bipartite graph $K_{m,n}$, grid graph $P_{m}\times P_{n}$, step grid graph $St_{n}$ and double step grid graph $DSt_{n}$ are absolute mean graceful graphs.

Keywords: Graphs, bipartite graph, absolute mean graceful labeling.

 Authors : J. Vijayasekhar and M. V. Subba Rao Title : Vibrational Frequencies of Water Isotopologues : Vibron Model Volume, Issue, Year : 5(4−E)(2017) Pages : 727−729

Abstract: In this paper, calculated vibrational frequencies of water isotopologues, Deuterium oxide ($D_{2}O$), Water-${}^{18}$O ($H_{2}{}^{18}O$) in fundamental level using Hamiltonian expression, which is in terms of invariant and Majorana operators describe stretching vibrations.

Keywords: Vibrational spectra, Water Isotopologues, Vibron Model.

 Authors : CH.Baby Rani, Sk.Mohiddin Shaw and G.Dharmaiah Title : Influence of Radiation on Heat and Mass Transfer in MHD Fluid Flow Over an Infinite Vertical Porous Surface with Chemical Reaction Volume, Issue, Year : 5(4−E)(2017) Pages : 731−740

Abstract: An examination is completed to think about the nonlinear MHD stream with heat and mass transfer attributes of an incompressible, viscous, electrically directing and Boussinesq fluid over a vertical moving permeable porous plate in presence of homogeneous chemical reaction of first request and heat radiation impacts. The problem is explained numerically utilizing the perturbation technique for the velocity, the temperature, and the concentration field. The impacts of different thermo-physical parameters on the velocity, temperature and concentration has been processed numerically and discussed about subjectively.

Keywords: Radiation, heat exchange, MHD, vertical plate, chemical reaction.

 Authors : S.Kannaki and L.Mary Louis Title : Normal-Rayleigh Stochastic Production Frontier Model Volume, Issue, Year : 5(4−E)(2017) Pages : 741−746

Abstract: In the production field, Stochastic Production Frontier Analysis (SPFA) plays a principal role in the measurement of technical efficiency. This paper presents the derivation of Normal-Rayleigh Stochastic Production Frontier Model (NRSPFM)for the estimation of technical efficiency. By using Maximum likelihood estimation method, the parameters of the model were evaluated.

Keywords: Rayleigh distribution, Normal-Rayleigh stochastic production frontier model, Technical efficiency, Maximum likelihood estimation.

 Authors : A.Mary Margaret and M.Trinita Pricilla Title : Neutrosophic Vague Generalized Pre-Closed Sets in Neutrosophic Vague Topological Spaces Volume, Issue, Year : 5(4−E)(2017) Pages : 747−759

Abstract: The aim of this paper is to introduce and develop a new class of sets namely neutrosophic vague generalized pre-closed sets in neutrosophic vague topological space. Further we have analyse the properties of neutrosophic vague generalized pre-open sets. Also some applications namely neutrosophic vague $T_{1/2}$ space, neutrosophic vague ${}_{p}T_{1/2}$ space and neutrosophic vague ${}_{gp}T_{1/2}$ space are introduced.

Keywords: Neutrosophic vague topological space, neutrosophic vague generalized pre-closed sets, neutrosophic vague $T_{1/2}$ space, neutrosophic vague ${}_{p}T_{1/2}$ space and neutrosophic vague ${}_{gp}T_{1/2}$ space.

 Authors : D. K. Thakkar and V. R. Dave Title : Edge Cover in a Hypergraph Volume, Issue, Year : 5(4−E)(2017) Pages : 761−768

Abstract: In this paper we introduce a new concept called an edge cover of a hypergraph. We characterize a minimal edge cover in hypergraph. We proved that if G is a hypergraph with minimum edge degree $\geq$ 2 then the complement of a minimal edge cover is an edge h-dominating set of G. We also proved a necessary \& sufficient condition under which the edge covering number of a hypergraph decreases when a vertex v is removed from the hypergraph. We also prove corresponding results for the partial sub hypergraph obtained by removing a vertex from the hypergraph.

Keywords: Hypergraph, Edge Dominating Set, Edge h-Dominating Set, Edge Cover, Minimal Edge Cover, Minimum Edge Cover, Edge Covering Number, Sub Hypergraph, Partial Sub Hypergraph.

 Authors : Kulkarni Sunita Jagannatharao and R. Murali Title : Rainbow Connection in Some Brick Product Graphs Volume, Issue, Year : 5(4−E)(2017) Pages : 769−774

Abstract: Let $G$ be a non-trivial connected graph on which is defined a colouring $c: E(G)\to \{1,2,3,\dots,k\}$, $k\in N$ of edges of $G$, where adjacent edges may be coloured the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are coloured the same. $G$ is rainbow-connected if it contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge colouring is called rainbow connection number of $G$, denoted by $r_{c}(G)$. In this paper we determine $r_{c}(G)$ for some brick product graphs $C(2n,m,r)$ associated with even cycles for $m=2$.

Keywords: Edge colouring, rainbow colouring, brick product graph.

 Authors : K. Sridevi, M.V.R. Kameswari and D.M.K Kiran Title : Common Fixed Point Theorems on Metric Space for Two and Four maps Using Generalized Altering Distance Functions in Five Variables and Applications to Integral Type Inequalities Volume, Issue, Year : 5(4−E)(2017) Pages : 775−791

Abstract: The concept of existence and uniqueness of fixed points by altering distance between points have been explored by many authors. In this paper, we obtain unique common fixed point results for two and four self mappings by altering distances sub compatible functions with generalization of contractive type condition and application to integral type inequalities.

Keywords: Altering distance functions, Sub compatible, Generalized altering distance, Integral type inequalities.

 Authors : Priya Baghel, V.H.Badshah, T.K.Jhala and Shivram Sharma Title : A Mathematical Model of Typhoid with Drug Resistance Effect Volume, Issue, Year : 5(4−E)(2017) Pages : 793−799

Abstract: In this paper, we consider a mathematical model of the type SIR (susceptible, infected and recovered) to understand the dynamic of the disease. We calculate the basic reproduction number $R_{0}$, using the next generation method, the disease free equilibrium and endemic equilibrium are established and their stability analysis done. We show that the disease free equilibrium point is globally asymptotically stable if $R_{o} <1$ and if $R_{o} >1$, there exist the endemic equilibrium state, which is also globally asymptotically stable.

Keywords: Typhoid, S. typhi, Multidrug Drug Resistance, basic reproduction number, Stability.

 Authors : B.N.R.Karuna, K.Lakshmi Narayan and B.Ravindra Reddy Title : Deterministic and Stochastic Dynamics of an Eco-epidemiological Model Volume, Issue, Year : 5(4−E)(2017) Pages : 801−808

Abstract: A three-compartmental eco-epidemiological model consisting of three species, namely, the susceptible prey, the infected prey and the predator population is considered. The SIR type deterministic epidemiological models are revisited and stochastic modeling for those models is introduced. The exponential and mean square stability for the stochastic model is derived and the considerable discrepancy between the results of stochastic and deterministic models is supported by numerical simulations with a hypothetical set of data.

Keywords: Eco-epidemiology, Stochastic stability, Predator-prey model, Lyapunov function.