Issue 1 Volume 6 (2018)

ISSUE 1− A ISSUE 1− B ISSUE 1− C ISSUE 1− D ISSUE 1− E ISSUE 1− Special Issue
Authors : V. Anusuya and P. Balasowandari
Title : Critical Path with Fuzzy Measure
Volume, Issue, Year : 6(1−D)(2018)
Pages : 603−609

Abstract: This paper presents the comparison on fuzzy critical path problem with trapezoidal type-2 fuzzy numbers. New ranking method and distance based similarity measure are introduced to identify the fuzzy critical path in a acyclic project network. An illustrative example is also included to demonstrate our proposed approach.

Keywords: Fuzzy critical path, distance based similarity measure, Trapezoidal type-2 fuzzy number, acyclic project network.

Authors : G. Sudha and R. Sophia Porchelvi
Title : A New Approach to Find Shortest Path in Intuitionistic Fuzzy Environment
Volume, Issue, Year : 6(1−D)(2018)
Pages : 611−616

Abstract: This paper presents to find the shortest path in a directed graph which an Triangular Intuitionistic Fuzzy Number, instead of a fuzzy number is assigned to each arc length. Procedures are designed to find the optimal path. An illustrative example is given to demonstrate the proposed approach.

Keywords: Shortest path, Triangular Intuitionistic Fuzzy Number (TIFN), Chen and Cheng Metric distance, Fuzzy network.

Authors : P. Gayathri and S. Pavithra
Title : A Consequential Computation of Degree-Based Topological Indices of Triangular Benzenoid
Volume, Issue, Year : 6(1−D)(2018)
Pages : 617−627

Abstract: There are various topological indices such as degree-based topological indices, distance based topological indices and counting related topological indices, etc. In this paper, we compute many degree-based topological indices like Randic, Geometric-Arithmetic, Sum-connectivity, Harmonic, First Zagreb, Second Zagreb, Second modified Zagreb, Inverse Sum, Alberston, Atom-Bond Connectivity, Symmetric-Division index and Augmented Zagreb indices for Triangular Benzenoid. Our results are extensions of many existing results.

Keywords: STopological index, Molecular graph, M-polynomial, Triangular Benzenoid.

Authors : Shailly Mahajan, Subash Kumar and Arun Kumar Tomer
Title : Comparison in Between Differential Transformation Method and Power Series Method for $13^{th}$ Order Differential Equation with Boundary Value Problems
Volume, Issue, Year : 6(1−D)(2018)
Pages : 629−634

Abstract: Differential transformation method is proposed to discover the solution of the higher order differential equations with the boundary value problem. The estimated solution of the problem is represented in the form of a rapid convergent series. A numerical example has been considered to demonstrate the effectiveness, exactness and implementation of the method and the results are compared with the exact solution. Afterward in the form of results are revealed graphically. The numerical result find by DTM are compared with the solution which are find by Power series Method and other presented method for instance the Variational Iteration Method (VIM) presented in this paper.

Keywords: Higher order Differential Equations, Differential Transformation Method, Power series Method, Taylor Series, Recurrence Relation and Variational Iteration Method.

Authors : Gagandeep Singh and Gurcharanjit Singh
Title : A Generalized Subclass of Close-to-Convex Functions
Volume, Issue, Year : 6(1−D)(2018)
Pages : 635−641

Abstract: The purpose of the present paper is to study a new generalized subclass of close-to-convex functions defined by subordination and we obtain coefficient estimates and also prove distortion and radii theorems for this class. Relevant connections of the results presented here with various known results are briefly indicated.

Keywords: Subordination, Univalent functions, Analytic functions, Convex functions, Close-to-convex functions.

Authors : K. Chandru and S. K. Narasimhamurthy
Title : Einstein Criterion for Finsler Space With Special $(\alpha, \beta)$-Metrics
Volume, Issue, Year : 6(1−D)(2018)
Pages : 643−649

Abstract: Einstein-Finsler metrics are very useful to study geometric structure of spacetime and to build applications in theory of relativity. In this paper, we consider the special $(\alpha, \beta)$-metric $L=\mu \frac{\alpha^2}{\beta}+\nu \frac{\beta^2}{\alpha}$ and obtained the Riemann curvature. Then we obtained the necessary and sufficient condition for that $(\alpha, \beta)$-metric to be Einstein metric, when $\beta$ is a constant killing form. Finally, we proved that the above metric is Einstein if and only if it is Ricci flat.

Keywords: $(\alpha,\beta)$-metrics, Riemannian curvature, Ricci curvature, Einstein Finsler space.

Authors : S. Jeyashri, S. Chandrasekar and M. Sathyabama
Title : Soft b$^{\#}$-open Sets and Soft b$^{\#}$-continuous Functions in Soft Topological Spaces
Volume, Issue, Year : 6(1−D)(2018)
Pages : 651−658

Abstract: Soft set theory was firstly introduced by Molodtsov in 1999 as a general Mathematical tool for dealing with uncertainty. He has shown several applications of this theory in solving many practical problems in Economics, Engineering,Social science, Medical science, etc. The purpose of this paper is to introduce a new type of soft open set is called Soft b$^{\#}$-open sets in Soft Topological spaces. Also, we discussed soft b$^{\#}$-continuous in Soft Topological spaces.

Keywords: Soft b$^{\#}$-open set, Soft b$^{\#}$-closed set, Soft b$^{\#}$-interior, Soft b$^{\#}$-closure, soft b$^{\#}$-continuous functions.

Authors : D. Muthuramakrishnan and G. Jayaraman
Title : Total Coloring of Splitting Graph of Path, Cycle and Star Graphs
Volume, Issue, Year : 6(1−D)(2018)
Pages : 659−664

Abstract: In 1965, the concept of total coloring was introduced by Behzad \cite{MB} and in 1967 he \cite{MG} came out new ideology that, the total chromatic number of complete graph and complete bi-partite graph. A total coloring of a graph $G$ is an assignment of colors to both the vertices and edges of $G,$ such that no two adjacent or incident vertices and edges of $G$ are assigned the same colors. In this paper, we have discussed the total coloring of splitting graph of Path, Cycle and Star graph. Also, we determine the total chromatic number of splitting graph of Path, Cycle and Star graphs.

Keywords: Path graph, Cycle graph, Star graph, Splitting graph, Total coloring and Total chromatic number.

Authors : Shikha Bansal
Title : Availability Analysis of a Repairable Redundant System Under Preemptive-Resume Repair Discipline
Volume, Issue, Year : 6(1−D)(2018)
Pages : 665−671

Abstract: This paper addresses the availability evaluation of a complex system comprising of two independent repairable subsystems in 1-out-of 2: F, under preemptive resume repair discipline. The failure rate follow exponential time distribution and repair time follow and general time distribution. The problem is formulated using the supplementary variable technique and various state probabilities have been obtained. A numerical example along with graphical representation has been appended to highlight the important conclusions.

Keywords: Redundancy, Supplementary variables technique, Laplace transform, Preemptive-Resume repair.

Authors : Dhir Singh and S. R. Singh
Title : Review of Literature and Survey of the Developed Inventory Models
Volume, Issue, Year : 6(1−D)(2018)
Pages : 673−685

Abstract: The review of the literature is an important feature of the research work that connects the past work with the current work. The literature on the inventory modelling is growing quickly. The goal of this article is to present a complete and updated information about the developed inventory models under different situations.

Keywords: Inventory models, deteriorating items, and trade credit.

Authors : M. I. Bosmia and K. K. Kanani
Title : Square Divisor Cordial Labeling in the Context of Vertex Switching
Volume, Issue, Year : 6(1−D)(2018)
Pages : 687−697

Abstract: A square divisor cordial labeling of a graph $G$ with vertex set $V(G)$ is a bijection $f$ from $V(G)$ to $\{1,2,\ldots,|V(G)|\}$ such that an edge $e=uv$ is assigned the label $1$ if $[f(u)]^2|f(v)$ or $[f(v)]^2|f(u)$ and the label $0$ otherwise, then $|e_f(0)-e_f(1)|\leq1$. A graph which admits square divisor cordial labeling is called a square divisor cordial graph. In this research article we prove that the graphs obtained by switching of a vertex in bistar, comb graph, crown and armed crown are square divisor cordial. In addition to this we also prove that the graphs obtained by switching of a vertex except apex vertex in helm and gear graph are square divisor cordial

Keywords: Square Divisor Cordial Labeling; Vertex Switching.

Authors : Sohan Tyagi
Title : Optimal Fuzzy Approach: Transportation Modelling
Volume, Issue, Year : 6(1−D)(2018)
Pages : 699−705

Abstract: The paper presents a classification and analysis of the results achieved using fuzzy logic to the model of transportation and traffic problems. In the recent years, research has been focused to estimating and techniques to help solve different engineering, management, control and computational problems. Natural systems teach us that very simple individual organisms can form system capable of performing highly tasks by dynamically interacting with each other. The main goal of the paper is to show how we can use fuzzy optimization approach when solving traffic and transportation. The fuzzy Ant system described in the paper represents an attempt to handle the uncertainty that sometimes exists in transportation problems. The potential application of the fuzzy Ant system in the field of traffic and transportation are discussed. The basic premises of fuzzy logic systems are presented as well as detailed analysis of fuzzy logic system developed to solve various traffic and transportation problems.

Keywords: Fuzzy logic, transportation modeling, Bee system, vagueness, traveling salesman problems, fuzzy Ant system.

Authors : Devender Kumar Kamboj, Dalip Singh and Vinod Kumar
Title : Countable KC Functions and Inversely Countable KC Functions
Volume, Issue, Year : 6(1−D)(2018)
Pages : 707−709

Abstract: In this paper, we introduce a new class of functions to be called countable KC and inversely countable KC functions. We study some relations between countable KC functions and inversely countable KC functions. We prove that if f is a continuous function from a space X to a Frechet, countable KC space Y, then $f(H)$ is closed in Y whenever H is a limit point compact set of X. It is also proved that if f is a countable KC function from a Frechet space X to a compact space Y, having closed point inverses, then f is continuous.

Keywords: Countable KC function, Inversely countable KC function, Continuous function, Frechet space.

Authors : Robinson Chelladurai and S. J. Maghy
Title : Graph Theory in Different Networks
Volume, Issue, Year : 6(1−D)(2018)
Pages : 711−717

Abstract: This paper gives a brief description of graph theory and networks .Various papers based on graph theory related to electrical network, communication network , biological network, transportation network, computer science applications have been studied and an overview has been presented here. More over author tries to find out the scope of graph theoretical concepts in further research.

Keywords: Bipartite graph, Tree, Cutset, Communication Network, Biological Network, Electrical Network.

Authors : P. R. Sharma and Ruchika Mehta
Title : Three-Dimensional Couette Flow Through Porous Medium Between Two Infinite Horizontal Parallel Porous Plates
Volume, Issue, Year : 6(1−D)(2018)
Pages : 719−730

Abstract: Aim of the paper is to investigate heat transfer through Couette flow of a viscous incrompessible fluid between two infinite horizontal parallel porous flat plates. The lower plate is stationary and the upper plate is moving with uniform velocity U subjected to transverse sinusoidal injection through lower plate and uniform suction through upper plate. Hence, the flow becomes three-dimensional due to fluid injection through lower plate. The governing equations of motion and energy are solved by regular perturbation technique and separation of variables technique. Velocity and temperature distributions are discussed numerically and shown through graphs. Nusselt number and coefficient of skin-friction at the plates are derived, discussed numerically and shown through graphs.

Keywords: Three-dimensionl, Couette flow, porous medium, skin-friction, Nusselt number.

Authors : S. Sasikala
Title : An Inventory Model for Deteriorating Items with Time Varying Demand and Shortages Using Fuzzy Environment
Volume, Issue, Year : 6(1−D)(2018)
Pages : 731−739

Abstract: Inventory in wider sense is defined as any idle resource of an enterprise. It is physical stock of goods, kept for the purpose of future affairs. Invention scheduling is a difficult task in industry with different operational constraints as well as strategic plan of the business revenue purpose and limit on finishing. To reduce the difficulties we introduce crisp and fuzzy model. Here fuzzy model consist of three methods gradient mean representation method, signed distance method, centroid method to deffuzify the total cost function with the help of triangular fuzzy number. The results obtained by these methods are compared with the help of a numerical example. Sensitivity analysis is also carried out to explore the effect of changes in the values of some of the system parameters. The aim of the study is to minimize the total cost function.

Keywords: Inventory, Deterioration, Shortages, Demand, Fuzzy variable, Triangular fuzzy number, Graded mean representation method, Signed distance method, Centroid method.

Authors : K. Arumugam, R. Abdul Saleem and S. Arunvasan
Title : Solving Fuzzy Assignment Problem using Symmetric Hexagonal Fuzzy Number
Volume, Issue, Year : 6(1−D)(2018)
Pages : 741−746

Abstract: Assignment problem is a special kind of Linear Programming Problem. In this paper, the cost values of the Fuzzy Assignment Problem are considered as Symmetric Hexagonal Fuzzy Numbers. First, the Symmetric Hexagonal Fuzzy Numbers are converted into crisp values using Hexagonal ranking method. Then the optimum assignment schedule of the Fuzzy Assignment Problem is obtained by usual Hungarian Method. The proposed approach is illustrated by a numerical example.

Keywords: Fuzzy Number, Fuzzy Set, Hexagonal Fuzzy Number, Symmetric Hexagonal Fuzzy Number, Fuzzy Assignment Problem, Hexagonal Ranking Function.

Authors : Akhilesh Jain, Rajesh Tokse, R. S. Chandel and Kamal Wadhwa
Title : Coupled Fixed Point Theorems in Ordered Non-Archimedean Intuitionistic Fuzzy Metric Space Using k-Monotone Property
Volume, Issue, Year : 6(1−D)(2018)
Pages : 747−755

Abstract: Michlelet [17] proved a theorem which assures the existence of a fixed point for fuzzy $\psi$-contractive mappings in the framework of complete non-Archimedean fuzzy metric spaces. Motivated by this, we introduce notion of k-monotone property in Intuitionistic fuzzy metric space and proved coupled fixed point theorems for the map satisfying the mixed monotone property in partially ordered complete non-Archimedean intuitionistic fuzzy metric space.

Keywords: Non- Archimedean property, k-monotone property, mixed monotone mappings, coupled fixed point, Fuzzy metric space, Intuitionistic Fuzzy metric space, Cauchy sequence, complete fuzzy metric space.

Authors : N. Elumalai, R. Manikandan and M. Kalaiselvi
Title : On Secondary k-Idempotent Bimatrices
Volume, Issue, Year : 6(1−D)(2018)
Pages : 757−762

Abstract: The concept of Secondary K-idempotent bimatrices are introduced. Some of characterization on secondary K-idempotent bimatrices are given.

Keywords: Idempotent matrix, idempotent bimatrix, S-k idempotent, S-k idempotent bimatrix, eigen value of S-k idempotent bimatrix.

Authors : Vikas Shinde
Title : Production System with Repair and Optimal Maintenance Policy
Volume, Issue, Year : 6(1−D)(2018)
Pages : 763−771

Abstract: The objective of this paper is to present the efficient and effective importance in productivity with the help of system repair and optimal maintenance. In fact the repair and maintenance is more concerning issue in the prevention of the losses due to breakdowns with in an enterprise. Queueing model is employed to analyze the system repair. Which describes the repair rate of the state is less or more. It is necessary to plan the activities to be carried out to ensure that the machines and plants of enterprises (production units) are working continuously. Therefore the gain is more important particularly when the production system grows as well as production quantity increases. Such study emphasis on optimal maintenance policy which is discussed in different cases. Availability of the system is examined by varying the various parameters.

Keywords: Optimal maintenance policy, Reliability and Availability function, Repair system, Queueing model.

Authors : K. Annie Kurien and M. S. Samuel
Title : Connectedness in Digital Spaces with the Khalimsky Topology
Volume, Issue, Year : 6(1−D)(2018)
Pages : 773−774

Abstract: In this paper we prove certain results related to connectedness in topological digital spaces.

Keywords: Digital spaces, Alexandroff space, Khalimsky topology, arc connected.

Authors : K. Annie Kurien and M. S. Samuel
Title : Relation between Khalimsky Topology and Slapal's Topology
Volume, Issue, Year : 6(1−D)(2018)
Pages : 775−779

Abstract: In this paper we study properties of both Khalimsky topology and Slapal's topology and the relation between them.

Keywords: Khalimsky topology, Slapal's topology, quotient topology, Alexandroff topological space, 4-adjacent, 8-adjacent.

Authors : M. R. Rajesh Kanna and R. Jagadeesh
Title : Distinct Energies of Paracetamol
Volume, Issue, Year : 6(1−D)(2018)
Pages : 781−789

Abstract: The concept of energy of a graph was introduced by I. Gutman in the year 1978. In this paper, we compute Energy, Siedel energy, Distance energy, Harary energy, Maximum degree energy, Randi\'{c} energy and Color energy of Paracetamol.

Keywords: Eigenvalues, Energy, Siedel Energy, Distance Energy, Harary Energy, Maximum Degree Energy, Randi\'{c} Energy, Color Energy, Paracetamol graph.

Authors : D. Senthilkumar and J. Meena
Title : Generalized k-Quasi-Hyponormal Composition Operators On Weighted Hardy Space
Volume, Issue, Year : 6(1−D)(2018)
Pages : 791−800

Abstract: In this paper we discuss the conditions for a composition operator and a weighted composition operator to be generalized k-quasi-hyponormal operator and also the characterization of generalized k-quasi-hyponormal composition operators on weighted Hardy space.

Keywords: Generalized k-quasi-hyponormal operator, composition operator, weighted Hardy space.

Authors : P. Rajalakshmi and S. Selvam
Title : Compactness in Smooth Bitopological Spaces
Volume, Issue, Year : 6(1−D)(2018)
Pages : 801−807

Abstract: The purpose of this paper is introducing the notion of smooth bitopological spaces. We define pairwise smooth-closure and pairwise smooth-interior of a fuzzy set and we study some of its properties. We give the definition for pairwise smooth compact set and show by an example that every pairwise smooth compact set need not be a smooth compact set. Finally we introduce the idea of pairwise smooth almost compact set and pairwise smooth nearly compact set.

Keywords: Smooth topological spaces, Smooth bitopological spaces, Pairwise smooth-closure, Pairwise smooth-interior, Pairwise smooth compact.

Authors : Hamid Shamsan, Fuad S. M. Alsarari and S. Latha
Title : Bounds for the Second $q$-Hankel Determinant of Certain Univalent Functions
Volume, Issue, Year : 6(1−D)(2018)
Pages : 809−817

Abstract: {We study the estimates for the Second $q$-Hankel determinant of analytic functions in a class which unifies a number of classes studied previously by Deepak Bansal, K. I. Noor, T. Yavuz, Sarika Verma, Shigeyoshi Owa and others. Our class includes $q$-convex, $q$-starlike functions and Ma-Minda starlike and convex functions.

Keywords: Second Hankel determinant, subordination, $q$-starlike and $q$-convex functions.

Authors : Ch. Santhi Sundar Raj, A. Natnael Teshale and U. M. Swamy
Title : Fuzzy Maximal Ideals of ADL's
Volume, Issue, Year : 6(1−D)(2018)
Pages : 819−824

Abstract: In this paper, we determine all maximal $L$-fuzzy ideals of a given ADL $A$ with a maximal element by establishing a one-to-one correspondence between maximal $L$-fuzzy ideals of $A$ and the pairs $(M,\alpha)$, where $M$ is a maximal ideal of $A$ and $\alpha$ is a dual atom in a complete lattice $L$ satisfying the infinite meet distributive law. Also, the notion of an $L$-fuzzy maximal ideal of $A$ is introduced and characterized.

Keywords: Almost Distributive Lattice (ADL); Maximal L-fuzzy ideal; L-fuzzy Maximal ideal; Complete Lattice.

Authors : M. R. Thirumagal and P. Murugadas
Title : Intuitionistic Fuzzy 2-(0- or 1-) Prime Ideals in Semirings
Volume, Issue, Year : 6(1−D)(2018)
Pages : 825−834

Abstract: The notion of intuitionistic fuzzy set was introduced by Atanassov as a generalization of the notion of fuzzy set. The purpose of this paper is to introduce the concept of three different types of intuitionistic fuzzy prime ideals and it has been shown that $A$ is an intuitionistic fuzzy 2-prime ideal of the semiring $S$ if and only if $A^C$ is an intuitionistic fuzzy $m_2$-system in $S$.

Keywords: Semiring, intuitionistic fuzzy ideal ($k$-ideal), intuitionistic fuzzy 2-(0- or 1-) prime ideal, intuitionistic fuzzy $m_2(m_0 ~or~m_1)$-systems.

Authors : K. Ponnalagu and P. Mounika
Title : A Study on Sequencing Problem in Two Different Environments
Volume, Issue, Year : 6(1−D)(2018)
Pages : 835−839

Abstract: To represent uncertainty the fuzzy numbers are ideally suited. In the present paper, we have considered the sequence performance measurements and job mean flow time as fuzzy in nature. Using fuzzy technological values, we consider fuzzy sequencing problem in a different method. The concepts of Average High Ranking Method (AHRM) are considered. This idea is used to minimize the total cost in earliness (lateness) at fuzzy environment. It has been shown that fuzzy approaches are used to tackle uncertainty into processing of jobs very effectively. In the procedure to obtain completion time, fuzzy ranking technique and defuzzification methods are involved to compare the fuzzy numbers and to facilitate the effective deterministic algorithm to get the best solution. In the present work, we have compared Crisp sequencing problem and Fuzzy sequencing problem. To support our study, numerical examples are illustrated in this paper.

Keywords: Crisp number, fuzzy number, triangular fuzzy number, sequencing problem, fuzzy environment.}

Authors : C. Rajesh Kumar
Title : $\mu$-$\alpha$-Semi Generalized Open Sets in Generalized Topological Spaces
Volume, Issue, Year : 6(1−D)(2018)
Pages : 841−845

Abstract: In this paper, I introduce a new class of sets in generalized topological spaces called $\mu$-$\alpha$-semi generalized open sets. Also I investigate some of their basic properties.

Keywords: Generalized topology, $\mu$-$\alpha$- open sets, $\mu$-$\alpha$-semi generalized open sets.