## Issue 1Volume 6 (2018)

ISSUE 4 − A ISSUE 4 − B ISSUE 4 − C
 Authors : S. Muthammai and P. Vidhya Title : Inverse Complementary Tree Domination Number of Graphs Volume, Issue, Year : 6(1−B)(2018) Pages : 205−208

Abstract: A non-empty set $D \subseteq V$ of a graph is a dominating set if every vertex in $V-D$ is adjacent to some vertex in $D$. The domination number $\gamma(G)$ of $G$ is the minimum cardinality taken over all the minimal dominating sets of $G$. A dominating set $D$ is called a complementary tree dominating set if the induced subgraph $\langle V-D\rangle$ is a tree. The complementary tree domination number $\gamma_{ctd}(G)$ of $G$ is the minimum cardinality taken over all minimal complementary tree dominating sets of $G$. Let $D$ be a minimum dominating set of $G$. If $V-D$ contains a dominating set $D^\prime$, then $D^\prime$ is called the inverse dominating set of $G$ w.r.t to $D$. The inverse domination number $\gamma^\prime(G)$ is the minimum cardinality taken over all the minimal inverse dominating sets of $G$. In this paper, we define the notion of inverse complementary tree domination in graphs. Some results on inverse complementary tree domination number are established, Nordhaus-Gaddum type results are also obtained for this new parameter.

Keywords: Dominating set, complementary tree dominating set, inverse complementary dominating set.

 Authors : Prem Pratap Vyas and Shashi Kant Title : Initial Coefficients Bounds for an Unified Class of Meromorphic Bi-univalent Functions Volume, Issue, Year : 6(1−B)(2018) Pages : 209−214

Abstract: In this paper, we introduce and investigate a new unified class of meromorphic bi-univalent functions defined on the domain $\{z\in\mathbb{C}: 1<|z|<\infty\}$, which are associated with meromorphic functions. We find estimates on the initial Taylor-MacLaurin coefficients for functions in these subclasses. Several new consequences of these results are also pointed out in the form of corollaries.

Keywords: Analytic function, Meromorphic function, Bi-Univalent functions, Coefficient estimates.

 Authors : K. P. R. Sastry, K. K. M. Sarma and S. Lakshmana Rao Title : A Fixed Point Theorem for Four Self Maps on a Multiplicative Metric Space and Consequences Volume, Issue, Year : 6(1−B)(2018) Pages : 215−221

Abstract: In this paper a fixed point theorem for a four self maps on a complete multiplicative metric space is proved and a fixed point theorem for six self maps is obtained as a corollary. Incidentally we obtain the result of Nisha Sharma et al. [8] as a corollary.

Keywords: Multiplicative metric space, Multiplicative contraction, Commuting mappings, Weakly compatible maps and Common fixed point.

 Authors : Sandeep Gupta Title : On Certain Sequence Space Volume, Issue, Year : 6(1−B)(2018) Pages : 223−227

Abstract: In this paper, we introduce a new sequence space $\ell_s^p, \, p \geq 1,$ which turns out to be an infinite dimensional separable Banach space in which H\"{o}lder inequality does not hold. It is shown that it is a proper subspace of $\ell^p$ to which $\ell^p$ is topologically equivalent. Apart from studying various algebraic and topological properties of $\ell_s^p$, its K\"{o}the- Toeplitz duals have also been computed.

Keywords: Sequence space, BK space, Schauder basis, K\"{o}the-Toeplitz duals.

 Authors : V. Sadhasivam, K. Logaarasi and T. Raja Title : On the Interval Oscillation of Impulsive Partial Differential Equations with Damping Term Volume, Issue, Year : 6(1−B)(2018) Pages : 229−242

Abstract: In this paper, we present some sufficient conditions for the oscillations of all solutions of impulsive partial differential equations. The results obtained here are based on the effect of impulses, delay and damping term in the sequence of subintervals of $\mathbb{R}_+$, which develops some well-known results for the equations without impulses, delay and the equations without damping term. Moreover, an example is presented to illustrate our main results.

Keywords: Oscillation, Impulse, Partial differential equations, Forcing term.

 Authors : Neeraj Anant Pande Title : Digits in Units Place of 2-Prime Factors Numbers Till 1 Trillion Volume, Issue, Year : 6(1−B)(2018) Pages : 243−247

Abstract: The first non-trivial type of ¡¥$k$-Prime Factors numbers¡¦ are 2-Prime Factors numbers. In this work, digits occurring in 2-Prime Factors numbers in units place are analyzed in thorough range of 1 trillion as well as within increasing ranges till 1 trillion for different block-sizes.

Keywords: Prime number, $k$-Prime Factors number, 2 Prime Factors number, Digits in units place.

 Authors : J. Tamilmani, D. Amsaveni and S. Meenakshi Title : Fuzzy Graph Cellular Spaces Volume, Issue, Year : 6(1−B)(2018) Pages : 249−254

Abstract: In the present work the initiation of fuzzy cellular graph and fuzzy graph cellular space are introduced. And fuzzy cellular graph and fuzzy $\mathcal{C}$ cellular are compared. Moreover some of the properties of fuzzy cellular graph are studied.

Keywords: Fuzzy cellular graph, fuzzy graph cellular space, fuzzy cellular $T_2$ space, fuzzy cellular regular space, fuzzy $\mathcal{C}$ cellular space.

 Authors : M. I. Qureshi and Sulakshana Bajaj Title : Two Hypergeometric Generating Relations Via Gould's Identity and Their Generalizations Volume, Issue, Year : 6(1−B)(2018) Pages : 255−269

Abstract: In the present paper, we have obtained hypergeometric generating relations associated with two hypergeometric polynomials of one variable $H_n^{(\alpha,\beta)}(x;m)$ and $\mathscr{B}_n^{(\alpha,\beta)}(x;m,\lambda,\mu)$ with their independent demonstrations via Gould's identity.As applications,some well known and new generating relations are deduced.Using bounded sequences, further generalizations of two main hypergeometric generating relations have also been given for two generalized polynomials $S_n^{(\alpha,\beta)}(x;m)$ and $T_n^{(\alpha,\beta)}(x;m,\lambda,\mu)$.

Keywords: Jacobi Polynomials, generalized Laguerre polynomial, generalized Rice polynomial of Khandekar, Gould's identity.

 Authors : M. R. Rajesh Kanna and R. Jagadeesh Title : Topological Indices of Vitamin A Volume, Issue, Year : 6(1−B)(2018) Pages : 271−279

Abstract: Graph theory has provided chemists with a variety of useful tools, such as topological indices. A topological index $Top(G)$ of a graph $G$ is a number with the property that for every graph $H$ isomorphic to $G$, $Top(H) =Top(G)$. In this paper, we compute $ABC$ index, $ABC_{4}$ index, Randi\'{c} connectivity index, Sum connectivity index, $GA$ index , $GA_{5}$ index, First \uppercase{Z}agreb index, Second \uppercase{Z}agreb index, First Multiple \uppercase{Z}agreb index, Second Multiple \uppercase{Z}agreb index, Augmented \uppercase{Z}agreb index, Harmonic index and Hyper \uppercase{Z}agreb index of Vitamin A.

Keywords: $ABC$ index, $ABC_{4}$ index, Randi\'{c} connectivity index, Sum connectivity index, $GA$ index , $GA_{5}$ index, First \uppercase{Z}agreb index, Second \uppercase{Z}agreb index, First Multiple \uppercase{Z}agreb index, Second Multiple \uppercase{Z}agreb index, Augmented \uppercase{Z}agreb index, Harmonic index, Hyper \uppercase{Z}agreb index and Vitamin A.

 Authors : V. Muthulakshmi and R. Manjuram Title : Interval Oscillation Criteria for Second-order Forced Delay Differential Equation with Riemann-Stieltjes Integral Volume, Issue, Year : 6(1−B)(2018) Pages : 281−295

Abstract: In this paper, we study the oscillatory behavior of second-order forced delay differential equation with Riemann-Stieltjes integral. By using the Riccati transformation technique, interval oscillation criteria of both El-Sayed type and Kong type are established, which generalize and extend some of the existing results. Finally, two examples are presented to illustrate the theoretical results.

Keywords: Interval criteria, Oscillation, Delay, Riemann-Stieltjes integral.

 Authors : K. R. Balasubramanian and V. Raja Title : Results on Intuitionistic Fuzzy k-ideals of Semiring Volume, Issue, Year : 6(1−B)(2018) Pages : 297−305

Abstract: In this paper, the product of intuitionistic fuzzy k-ideals is introduced. Also some basic properties are derived. The relationship between intuitionistic fuzzy k-ideal $A, B$ and $A\times B$ are proposed. Some theorems related to the above concepts are stated and proved.

Keywords: Semi-ring, Intuitionistic Fuzzy k-ideal.

 Authors : T. J. Rajesh Kumar and T. K. Mathew Varkey Title : Gaussian Prime Labeling of Some Product Graphs Volume, Issue, Year : 6(1−B)(2018) Pages : 307−311

Abstract: A graph G on vertices is said to have prime labeling if there exists a labeling from the vertices of G to the first n natural numbers such that any two adjacent vertices have relatively prime labels.Gaussian integers are the complex numbers of the form $a+bi$ where $a,b \in \mathscr{Z}$ and $i^2 = -1$ and it is denoted by $\mathscr{Z}[i]$. A Gaussian prime labeling on G is a bijection from the vertices of G to $[\psi_n]$, the set of the first n Gaussian integers in the spiral ordering such that if $uv \in E(G)$, then $\psi_{(u)}$ and $\psi_{(v)}$ are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of product graphs.

Keywords: Gaussian integers, Gaussian prime labeling, product graphs.

 Authors : Purushottam Jha and Manisha Shrivastava Title : A New Generalized Continuity by Using Generalized-Closure Operator Volume, Issue, Year : 6(1−B)(2018) Pages : 313−324

Abstract: We introduce a new class of generalized closed sets namely, $_{N}D_{\beta}$--closed sets, which is weaker than g--closed(generalized--closed) \cite{lev70}, $semi^{\ast}$--closed sets \cite{rob12}, $pre^{\ast}$--closed sets \cite{sel12} and $D_{\alpha}$--closed sets \cite{say16} in topological spaces. Moreover, we introduce $_{N}D_{\beta}$--continuous and $_{N}D_{\beta}$--irresolute functions and study its fundamental properties.

Keywords: $_{N}D_{\beta}$--closed, $_{N}D_{\beta}$--neighborhood, $_{N}D_{\beta}$--limit point, $_{N}D_{\beta}$--continuous, $_{N}D_{\beta}$--irresolute.

 Authors : N. K. Sahoo and P. K. Tripathy Title : An EOQ Model for Quadratic Demand Rate, Parabolic Deterioration, Time Dependent Holding Cost with Partial Backlogging Volume, Issue, Year : 6(1−B)(2018) Pages : 325−332

Abstract: An economic order quantity model has been developed for deteriorating items with demand rate as a quadratic function of time. In the model, deterioration rate and holding cost are time dependent. The inventory shortage is discussed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. Results are illustrated with numerical example along with sensitivity analysis for the model with respect to various parameters is carried out.

Keywords: Quadratic Demand Rate, Deterioration Rate, Partial Backlogging.

 Authors : E. N. Satheesh Title : Roman Labeling in Digraphs Volume, Issue, Year : 6(1−B)(2018) Pages : 333−336

Abstract: Let G be a connected graph. Roman labeling of G is a function $f: V(G) \to \{0, 1, 2\}$ such that any vertex with label 0 is adjacent to a vertex with label 2. The function value $f(v)$ of a vertex v of the graph G is called the label of v. Weight of a Roman labeling, f is defined as the sum of all vertex labels.That is, $\mathrm{w}\left(f\right)=\sum\limits_{v\in V(G)}f(v)$. Roman number of a graph G is defined as the minimum weight of a Roman labeling on G and is denoted by $S(G)$. It was introduced by Satheesh and Suresh Kumar [3]. In this paper, we extend the idea of Roman labeling and Roman number to Digraphs.

Keywords: Roman labeling, Roman number, Digraphs.

 Authors : M. Anitha and S. Balamurugan Title : Strong Efficient Open Domination in Graphs Volume, Issue, Year : 6(1−B)(2018) Pages : 337−342

Abstract: Let $G=(V, E)$ be a simple, undirected, finite graph without isolated vertices. A subset $D$ of $V(G)$ is a {\it Strong(Weak) efficient open dominating set} of $G$ if $\left|N_s(v)\cap D\right|=1$ $(\left|N_w(v)\cap D\right|=1)$, for every $v\in V(G)$ where $N_s(v)$ and $N_w(v)$ are strong and weak neighborhood respectively. The minimum cardinality of $D$ is called as {\it strong(weak) efficient open domination number} and is denoted by $\gamma_{ste}(G)$ $(\gamma_{wte}(G))$ of $G$. A graph $G$ is {\it strong(weak) efficient open dominating graph} if it contains a strong(weak) efficient open dominating set. We write a program to check whether the given graph is strong efficient open dominatable or not in C Language.

Keywords: Strong efficient open dominating set, Strong efficient open dominating graph, Strong efficient open domination Number, Weak efficient open dominating set, Weak efficient open domination number.

 Authors : A. Mary Priya Dharsini and U. Karuppiah Title : Best Proximity Points for Multiplicative Modified Rational Proximal Contraction Mapping on Multiplicative Metric Spaces Volume, Issue, Year : 6(1−B)(2018) Pages : 343−349

Abstract: In this paper, we prove best proximity point theorems in multiplicative metric spaces satisfying multiplicative modified rational proximal contraction condition of the first kind. The presented results extend, generalize and improve some known results from best proximity point theory. We also give some examples to illustrate and validate our definitions and results.

Keywords: Best proximity points, fixed points, multiplicative metric spaces, multiplicative modified rational proximal contraction mappings of the first kind.

 Authors : A.Kumar, R.S.Chandel, R.Shrivastava, K.Shrivastava and S.Kumar Title : A Mathematical Model for Effect of Couple-stress in Fluid Through the Blood Vessel with Mild Stenosis Volume, Issue, Year : 6(1−B)(2018) Pages : 351−358

Abstract: A mathematical model is developed to study the effect of couple stress on the blood flow through two-dimensional symmetric arteries with stenosis. The present model is consists of a core region to be a couple stress fluid and the peripheral layer of Newtonian fluid. Analytical expressions have been obtained for resistance to flow and wall shear stress with the height of stenosis. The numerical values are derived from these analytic expressions. The effect of various parameters on these flows has been examined and depicted through graphs for different values. The information of blood flow could be very useful in the development of new diagnosis tool for the arterial disease.

Keywords: Physiological flow, stenosis, couple stress fluid, peripheral layer.

 Authors : K.P.V. Preethi, M. Chitra Devi, R. Swaminathan and R. Poovazhaki Title : The New Homotopy Perturbation Method (NHPM) for Nonlinear Parabolic Equation in Chemical Sciences Volume, Issue, Year : 6(1−B)(2018) Pages : 359−367

Abstract: In this paper, we have presented new homotopy perturbation method (NHPM) to solve the system of nonlinear parabolic equation in chemical sciences. The NHPM allows for the solution of the nonlinear parabolic equations to be calculated in the form of a series with easily computable term by converting a non-linear differential equation into linear differential equation. The obtained results are presented and only one iteration is required to obtain an approximate solution that is accurate and efficient. Simple analytical expressions for the concentration of substrate and product have been derived for all values of reaction parameters using the new homotopy perturbation method (NHPM). The analytical results are also compared with numerical ones and a good agreement is obtained.

Keywords: Nonlinear equation, New homotopy perturbation method, Parabolic equation.

 Authors : Jitendra Awasthi Title : A New Class of Meromorphic Multivalent Functions Involving an Extended Linear Derivative Operator of Ruscheweyh Volume, Issue, Year : 6(1−B)(2018) Pages : 369−375

Abstract: In this paper,we introduce and discuss a new subclass $A_{p}(\lambda ,\alpha )$ of meromorphic multivalent functions in the punctured unit disk $U^{\ast }=\{z:z$ $\epsilon$ $C,0<\left\vert z\right\vert <1\}$ defined by a certain extended linear derivative operator of Ruscheweyh. Coefficients inequality,distortion theorems,closure theorem for this class are obtained.We also obtain radius of starlikeness and radius of convexity for this class.

Keywords: Meromorphic Multivalent Functions, Ruscheweyh Derivatives, Meromorphic Starlike Functions.

 Authors : G. B. Antony Xavier, S. J. Borg, S. Jaraldpushparaj and C. Sowmiya Title : Discrete $q$-heat Equation Model with Shift Values Volume, Issue, Year : 6(1−B)(2018) Pages : 369−375

Abstract: In this paper, we investigate the generalized $q$-partial difference operator and propose a model of it in discrete q-heat equation with shift values. The diffusion of heat is studied in dimensions up to three and several solutions are postulated for the same. Through numerical simulations using MATLAB, solutions are validated and applications are derived.

Keywords: Generalized partial difference equation, partial $q$-difference operator and $q$-heat equation.

 Authors : V. J. Kaneria, Om Teraiya and Parinda Bhatt Title : Generalized Odd-Even Sum Labeling and Some $\alpha-$Odd-Even Sum Graphs Volume, Issue, Year : 6(1−B)(2018) Pages : 381−385

Abstract: A $(p,q)$ graph $G$ is said to be an $\alpha$-odd-even sum graph if it admits an odd-even sum labeling $f$ defined by Monika and Murugan \cite{9} by adding an addition condition that there is a positive integer $k{(0 Keywords:$\alpha$-odd-even sum labeling, Grid graph, Step grid graph, Splitting graph.  Authors : Mahesh Chalpuri and J. Sucharitha Title : Approximation Weights of Gauss Quadrature Method Volume, Issue, Year : 6(1−B)(2018) Pages : 387−393 Abstract: To find the exact area of Definite Integral of continuous function on the Interval [$a, b$] is very problematic. In Numerical methods, the most popular method to find the area of finite Definite Integral is Gauss Legendre Quadrature Method (GLQM). In this GLQM, the weights are very difficult to find. In this paper, the new method is obtained using the New Weights which are nearest to GLQM weights. Also, the order of GLQM depends on the number of nodes, whereas the order of this new method is always two. Keywords: Gauss Quadrature method, Numerical method, Newton-Cotes method, Numerical Integration, Lobatto quadrature method.  Authors : T. Indira and S. Vijayalakshmi Title :$I_{\ddot{g}}$-Separation Axioms in Ideal Topological Spaces Volume, Issue, Year : 6(1−B)(2018) Pages : 395−401 Abstract: This paper mainly concentrates on the study of separation axioms in Ideal Topological spaces. In particular, we delibrate the properties of I$_{\ddot{g}}-T_{0}$space, I$_{\ddot{g}}-T_{1}$space, I$_{\ddot{g}}-T_{2}$space, I$_{\ddot{g}}-Q_{1}$space and I$_{\ddot{g}}-Q_{2}$space. Keywords:$I_{\ddot{g}}-T_{0}$space,$I_{\ddot{g}}-T_{1}$space,$I_{\ddot{g}}-T_{2}$space,$I_{\ddot{g}}-Q_{1}$space and$I_{\ddot{g}}-Q_{2}\$ space.

 Authors : Wei-Kai Lai Title : Nesbitt Type Inequalities Volume, Issue, Year : 6(1−B)(2018) Pages : 403−406

Abstract: In 2009 Wei and Wu introduced an n-variable version of Nesbitt's inequality. In this paper we provide a different proof using Radon's inequality. We also use the same technique to prove several of its variations.