## Issue 1Volume 6 (2018)

ISSUE 1− A ISSUE 1− B ISSUE 1− C ISSUE 1− D ISSUE 1− E ISSUE 1− Special Issue
 Authors : K. S. Aboodh, R. A. Farah, I. A. Almardy and F. A. ALmostafa Title : Some Application of Aboodh Transform to First Order Constant Coefficients Complex Equations Volume, Issue, Year : 6(1−A)(2018) Pages : 1−6

Abstract: In this work, we present a reliable Aboodh transform method to solve first order constant coefficients complex equation. This method provides an effective and efficient way of solving a wide range of linear operator equations.

Keywords: Aboodh Transform, first order constant coefficients complex equation, linear operator equations.

 Authors : G. Nirmala and G. Uthra Title : AHP Based on Trapezoidal Linguistic Fuzzy Preference Relation Volume, Issue, Year : 6(1−A)(2018) Pages : 7−14

Abstract: Consistency is an important issue in any decision making problem. Lack of consistency yields to unreliable solution. This paper proposes Trapezoidal fuzzy linguistic preference relation to improve the consistency of Fuzzy analytic Hierarchy process. In this study using Trapezoidal fuzzy linguistic preference relation and using only n-1 pairwise comparisons, a pairwise comparison matrix is constructed. The matrix satisfies additive reciprocal property and consistency. The proposed method is applied to the example given in \cite{14} and the results are compared. This study provides the same ranking as in \cite{14} without much involvement of decision maker.

Keywords: Fuzzy AHP, consistency, Trapezoidal fuzzy number, linguisticvariable, Decision making.

 Authors : L. Vidyarani and M. Vigneshwaran Title : Quasi Supra N-closed Map and Supra N-normal Space Volume, Issue, Year : 6(1−A)(2018) Pages : 15−22

Abstract: The purpose of this paper is to give a new type of map called Quasi supra N-closed map and we obtain its basic properties and also we introduce the concept of supra N-normal spaces and study some fundamental properties.

Keywords: Quasi supra N-closed map, Quasi supra N-open map, supra N-normal space, weakly supra N-normal space.

 Authors : A. Vadivel and C. Sivashanmugaraja Title : On $\gamma$-regular Semi-open Sets Volume, Issue, Year : 6(1−A)(2018) Pages : 23−28

Abstract: We introduce the concepts of $\gamma$-regular semi-open set, $\gamma_{rs}$-set, $\gamma^{rs}$-set, generalized $\gamma_{rs}$-set, generalized $\gamma^{rs}$-set, regular semi-$T_{1}^{\gamma}$ space and regular semi-$R_{0}^{\gamma}$ space by using $\gamma$-regular open sets.

Keywords: Generalized topology, generalized open sets, $\gamma$-open sets, $\gamma$-regular semi-open sets, regular semi-$T_{1}^{\gamma}$ space, regular semi-$R_{0}^{\gamma}$ space.

 Authors : Tara Dhakate, Vinod Varghese and Lalsingh Khalsa Title : An Analytical Solution for Transient Asymmetric Heat Conduction in a Multilayer Elliptic Annulus and its Associated Thermal Stresses Volume, Issue, Year : 6(1−A)(2018) Pages : 29−42

Abstract: A closed-form solution is presented for the transient heat conduction problem in the elliptic annulus with multiple layers in the radial direction, but time-independent volumetric heat sources in each layer. The solution is obtained applying an integral transform technique analogous to Vodicka's approach considering series expansion function in terms of an eigenfunction to solve the heat conduction partial differential equation in elliptical coordinates. The results are obtained as a series solution in terms of Mathieu functions and hold convergence test. The temperature and thermal stresses for the three-layer elliptic region have been computed numerically and exhibited in a graphical form, by using dimensionless values and therefore discussed.

Keywords: Multilayer elliptic annulus, heat conduction, integral transform, thermal stresses.

 Authors : M. Dube and Jagriti Sharma Title : Inflection Point Control by Using Rational Cubic Spline Volume, Issue, Year : 6(1−A)(2018) Pages : 43−50

Abstract: Consider the rational spline of the form cubic / quadratic with two shape parameters. The shape control of rational interpolation such as value point control, inflection point control and convexity control at a point is studied. The main aim of the paper is to control the shape of interpolating curve. Also, the numerical examples are given to validate the proposed interpolation technique.

Keywords: Rational Spline, Value Control, Inflection-point Control, Convexity Control.

 Authors : S. Sudhakar, V. Maheshwari and V. Balaji Title : Cordial Labeling for Star Graphs Volume, Issue, Year : 6(1−A)(2018) Pages : 51−54

Abstract: Cordial labeling is used to label the vertices and edges of a graph with $\left\{ 0,1 \right\}$ under constraint, such that the number of vertices with label $0$ and $1$ differ by atmost $1$ and the number of edges with label $1$ and $0$ differ by atmost $1$. In this paper we prove that the two star graph $K_{1,m} \wedge K_{1,n}$ with a wedge in common is a cordial graph for all $m\geq1$ and $n\geq1$.

Keywords: Cordial graph, Star graph, Cordial labeling.

 Authors : G. Somashekhara, S. Girish Babu and N. Pavani Title : On a Semi-symmetric Metric Connection in Generalized Sasakian Space forms Satisfying Certain Curvature Conditions Volume, Issue, Year : 6(1−A)(2018) Pages : 55−62

Abstract: In this paper,we investigate Ricci pseudo-symmetric and Ricci generalized pseudo-symmetric semi-symmetric metric connection in generalized sasakian space forms satisfying the curvature condition $\tilde{S}.\tilde{R}=0$.

Keywords: Generalized Sasakian Space, Curvature Conditions, Semi-symmetric Metric Connection.

 Authors : B. Shanmugapriya, K. Ramesh and S. Chitra Title : $IF\alpha^{**}g$ Closed Sets in Intuitionistic Fuzzy Topological Spaces Volume, Issue, Year : 6(1−A)(2018) Pages : 63−71

Abstract: In this paper, we introduce and study new classes of sets called $IF\alpha^{**}g$ closed set in intuitionistic fuzzy topological spaces and $IF\alpha^{**}g$ open set in intuitionistic fuzzy topological spaces. We focus upon the some of their basic properties.

Keywords: Intuitionistic fuzzy topology, $IF\alpha^{**}g$ closed set in intuitionistic fuzzy topological spaces, $IF\alpha^{**}g$ open set in intuitionistic fuzzy topological spaces.

 Authors : R. L. Patel, S. K. Pandey and R. N. Singh Title : Ricci Solitons on ($\epsilon$)-Para Sasakian Manifolds Volume, Issue, Year : 6(1−A)(2018) Pages : 73−82

Abstract: The object of the present paper is to study some semi-symmetry conditions on ($\epsilon$)-para Sasakian manifolds admitting Ricci solitons.

Keywords: ($\epsilon$)-para Sasakian manifolds, Ricci solitons, projective curvature tensor, space-like vector field, light-like vector field.

 Authors : G. N. Pathan and P. H. Bhathawala Title : A Mathematical Model for Unemployment Control-An Analysis with and without Delay Volume, Issue, Year : 6(1−A)(2018) Pages : 83−93

Abstract: The paper presented a mathematical model to control unemployment using dynamic system. In this process we consider the situation where job search is open for native unemployed and new migrant workers enter into the territory as unemployed. So, government of that territory feels more burden of unemployment and make efforts for creating new vacancies with the help of private sector. Also, we assume that native unemployed and new migrant workers both make efforts for self-employment. We studied the local stability of the non-negative equilibrium point with and without delay. Numerical simulation has been carried out to illustrate the analytical result.

Keywords: Employed persons, unemployed persons, self-employment, migration, delay, dynamic variables.

 Authors : I. Annammal Arputhamary and M. Helda Mercy Title : Strong Edge Coloring of Some Classes of Unicyclic Graphs Volume, Issue, Year : 6(1−A)(2018) Pages : 95−98

Abstract: Let \textit{G } be an undirected simple graph. A strong edge coloring of a graph \textit{G } is a function $f : E \to \{1, 2,\dots, k\}$ such that $f(e_{1} )\ne f(e_{2} )$ whenever $e_{1}$ and $e_{2}$ lie within distance 2 from each other. In other words, no two edges lie on a path of length 3 receive same colors. The smallest number of colors essential for strong edge coloring of a graph \textit{G } is entitled as strong chromatic index and is represented by $\chi '_{s} (G)$. In this paper, we investigate strong chromatic index of some classes of unicyclic graphs.

Keywords: Strong edge coloring, strong chromatic index, unicyclic graph.

 Authors : Garima Gadkari, M. S. Rathore and Naval Singh Title : Some Fixed Point Theorems for a Pair of Mappings Under the Rational Inequality in Complex Valued b-Metric Space Volume, Issue, Year : 6(1−A)(2018) Pages : 99−106

Abstract: In this problem we improve a common fixed point theorem for a pair of mappings under the rational inequality and obtain sufficient conditions for existence and uniqueness in complex valued b-metric spaces.

Keywords: Common fixed point, Complex valued b-metric space, Cauchy sequence, Coincidence point, Weakly compatible mappings.

 Authors : A. Delman, S. Koilraj and P. Lawrence Rozario Raj Title : SD Prime Cordial Labeling of Some Special Graphs Volume, Issue, Year : 6(1−A)(2018) Pages : 107−114

Abstract: In this paper we investigate the SD - prime cordial labeling for $G \cup (P_n \odot K_1), G \cup K_{1,n,n}, G \cup PS_n$ and $G \cup P_n$.

Keywords: Prime graph, SD prime graph, SD prime cordial graphs.

 Authors : J. Periaswamy and N. Selvi Title : Geometric Mean Labeling of Union and Product of Some Standard Graphs with Zero Divisor Graphs Volume, Issue, Year : 6(1−A)(2018) Pages : 115−120

Abstract: In this paper, Geometric mean labeling of $C_m\cup \Gamma(Z_n)$, $mC_n\cup \Gamma(Z_n)$, $nC_3\cup \Gamma(Z_n)$, $n\Gamma(Z_{25})$, $n\Gamma(Z_{25})\cup P_m$, $n\Gamma(Z_{25}) \cup C_m$, $n\Gamma(Z_{25})\cup mC_k$, $\Gamma(Z_{2p})\cup \Gamma(Z_{2p})$, $P_m\times \Gamma(Z_8)$ and $P_n\times \Gamma(Z_9)$ are investigated.

Keywords: Graph Lebeling, Geometric Mean Labeling.

 Authors : M. Lellis Thivagar, V. Antonysamy and M. Arockia Dasan Title : Revelation of Nano Topology in $\check{C}$ech Rough Closure Spaces Volume, Issue, Year : 6(1−A)(2018) Pages : 121−131

Abstract: The purpose of this paper is to derive Nano topology in terms of $\check{C}$ech rough closure operators. In addition to this, we also establish the continuous functions on $\check{C}$ech rough closure space and its properties. Finally from these, we evolve a $\check{C}$ech nano topological space that satisfies the topological axioms on infinite universe.

Keywords: Rough sets, $\check{C}$ech rough closure operators, $\check{C}$ech rough closure spaces, $\check{C}$ech rough continuity, $\check{C}$ech nano topological spaces.

 Authors : J. Periaswamy and N. Selvi Title : Edge Sum Index of a Graph in a Commutative Ring Volume, Issue, Year : 6(1−A)(2018) Pages : 121−131

Abstract: Let $\Gamma(Z_{n})$ be a graph. A bijection $f:E(\Gamma(Z_{n})) \to Z^{+}$, where $Z^{+}$ is a set of positive integers is called an edge mapping of the graph $\Gamma(Z_{n})$. Now, we define, $F(v)=\Sigma\{f(e);$ e is incident on $v \}$ on $V(\Gamma(Z_{n}))$. Then, F is called the edge sum mapping of the edge mapping f. $\Gamma(Z_{n})$ is said to be an edge sum graph if there exists an edge mapping $f:E(\Gamma(Z_{n})) \to N^{+}$ such that f and its corresponding edge sum mapping. F on $V(\Gamma(Z_{n}))$ satisfy the following conditions: (i) F is into mapping to$Z^{+}$. That is, $F(v) \in Z^{+}$, for every $v \in E(\Gamma(Z_{n}))$. (ii) If $e_{1},e_{2},\dots,e_{n} \in E(\Gamma(Z_{n}))$ such that $f(e_{1})+f(e_{2})+\dots f(e_{n}) \in Z^{+}$, then $e_{1},e_{2},\dots,e_{n}$ are incident on a vertex in $\Gamma(Z_{n})$. In this paper, we evaluated the edge sum index of some standard graphs in zero divisor graph.

Keywords: Graph Lebeling, Edge Sum Index of a Graph, Commutative Ring.

 Authors : U. M. Prajapati and R. M. Gajjar Title : Cordiality in the Context of Duplication in Helm and Closed Helm Volume, Issue, Year : 6(1−A)(2018) Pages : 139−145

Abstract: Let $G = (V(G) , E(G))$ be a graph and let $\displaystyle f:V(G)\rightarrow \{0,1\}$ be a mapping from the set of vertices to \{0,1\} and for each edge $uv \in E$ assign the label $|f(u)-f(v)|$. If the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labled with 0 and the number of edges labeled with 1 differ by at most 1, then $f$ is called a cordial labeling. We discuss cordial labeling of graphs obtained from duplication of certain graph elements in helm and closed helm.

Keywords: Graph Labeling, Cordial Labeling, Cordial Graph.

 Authors : S. Priyadharsini, P. Gowri and T. Aparna Title : Solution of Fractional Telegraph Equation with Fuzzy Initial Condition Volume, Issue, Year : 6(1−A)(2018) Pages : 147−154

Abstract: In this work, we analyze the method of finding the solution of fractional telegraph equation of the form $\frac{\partial^{2 \alpha} u}{\partial x^{2 \alpha}}+a \odot \frac{\partial^\alpha u}{\partial t^\alpha} +b \odot u = \frac{\partial^2 u}{\partial x^2},$ with fuzzy initial condition, where $^C\! D$ is a Caputo fractional derivative. Comparison has been done between integer order and fractional order model. Some numerical illustrations are provided to explain the proposed theory.

Keywords: Fractional Telegraph equations, fuzzy differential equations, Laplace Transform.

 Authors : Nidhi Jolly, Priyanka Harjule and Rashmi Jain Title : On the Solution of General Family of Fractional Differential Equation Involving Hilfer Derivative Operator and $\overline{H}$-function Volume, Issue, Year : 6(1−A)(2018) Pages : 155−162

Abstract: In this paper, we first give solution to a general family of fractional differential equation involving Hilfer derivative operator and the fractional integral operator whose kernel is the $\overline{H}$-function. Next, we record here solutions of two fractional differential equations involving the function associated with Gaussian Model free energy and Polylogarithm function of order g as special cases of our main result. These special cases are believed to be new. On account of the general nature of\, $\overline{H}$-function in our main findings, the results derived earlier by Srivastava et al. \cite{47}, Srivastava and Tomovski \cite{11} and Tomovski et al. \cite{9} follow as special cases.

Keywords: $\overline{H}$-Function, Laplace Transform, Mittag-Leffler function, Fractional Integral Operator, General Family of Fractional Differential Equation.

 Authors : Jyoti Gupta and M. Shrivastava Title : Intuitionistic Fuzzy-$(\rho, \sigma)$-Semi Pre Compact Spaces Volume, Issue, Year : 6(1−A)(2018) Pages : 163−170

Abstract: In the present paper, we define and study variety of intuitionistic fuzzy (or IF-) compact spaces: IF-alpha compact space, IF-semi compact space, IF-pre compact space and IF-semi pre compact spaces in sense of Sostak. Further we investigate their significant properties and relationship among them.

Keywords: Intuitionistic fuzzy sets, intuitionistic fuzzy topological spaces, intuitionistic fuzzy continuous map, intuitionistic gradation of openness, intuitionistic fuzzy compactness.

 Authors : Vinayak Eknath Nikumbh Title : Fuzzy Topological Fuzzy Ordered Spaces Volume, Issue, Year : 6(1−A)(2018) Pages : 171−177

Abstract: In this paper by combining the notions of certain types of fuzzy order and fuzzy topology we introduce fuzzy topological fuzzy ordered spaces. Its various properties are analyzed. We also develop and study order separation axioms called $fTi$ separation axioms for `fuzzy topological fuzzy ordered spaces'. The relationships between some of these $fTi$ separation axioms are also studied.

Keywords: Fuzzy order, fuzzy topology, separation axioms.

 Authors : Pablo I. Pucheta Title : On The k-Caputo-Fabrizio Fractional Derivative and its Applications Volume, Issue, Year : 6(1−A)(2018) Pages : 179−185

Abstract: In this paper a generalization of the derivative due to Caputo and Fabrizio in \cite{3} is introduced. We present some useful properties, evaluate its Laplace transform and also obtain the k-fractional integral associated with the new fractional derivative. We will also resolving the k-fractional logistic equation Given by Cerutti \cite{4} with a new fractional operator called on the k-Caputo-Fabrizio´s fractional derivative with a non-singular kernel. In the same way we will see that when $k=a=1$ the solution matches with the one given by Camargo and Bruno-Alfonzo \cite{6}.

Keywords: Fractional Calculus, Laplace Transform, k-Caputo-Fabrizio fractional derivative.

 Authors : A. A. Wani, P. Catarino and R Ur Rafiq Title : On the Properties of $k$-Fibonacci-Like Sequence Volume, Issue, Year : 6(1−A)(2018) Pages : 187−198

Abstract: In this article, we introduce a new generalization of Fibonacci sequence and we call it as $k$-Fibonacci-Like sequence. After that we obtain some fundamental properties for $k$-Fibonacci-Like sequence and also we present some relations among $k$-Fibonacci-Like sequence, $k$-Fibonacci sequence and $k$-Lucas sequence by some algebraic methods.

Keywords: Fibonacci Sequence, Generalized Fibonacci Sequence, $k$-Fibonacci Sequence and $k$-Lucas Sequence.

 Authors : J. Beny and J. Maria Joseph Title : Best Proximity Point Theorems for Some Well Known Mappings in Complete $b-$Metric spaces Volume, Issue, Year : 6(1−A)(2018) Pages : 199−204

Abstract: In this paper, we prove best proximity point theorems for types of cyclic $b-$contraction mappings in the setting of complete $b-$ metric spaces which generalize some results in the current literature.

Keywords: Fixed point, best proximity point, complete metric space, contraction, $b-$metric space.