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Issue 2
Volume 6 (2018)

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Authors |
: | Shakir Msahir Atshan |

Title |
: | Monotone Iterative Technique for Nonlinear Volterra-Fredholm Integro-Differential Equations |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 1−9 |

**Abstract: ** This paper presents an efficient algorithm based on a monotone method for the solution of a class of nonlinear Volterra-Fredholm integro-differential equations of second order. This method is applied to derive two monotone sequences of upper and lower solutions which are uniformly convergent. Theoretical analysis of the existence and convergence of those sequences are discussed. The numerical results demonstrate reliability and efficiency of the proposed technique.

**Keywords: ** Monotone method, Lower and upper solutions, Volterra-Fredholm integro-differential equation.

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Authors |
: | V. Chinnadurai and K. Arulmozhi |

Title |
: | Some Characterization of Fuzzy Soft $\Gamma$-Semigroups |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 11−18 |

**Abstract: ** The purpose of this paper is to introduce the notion of fuzzy soft prime ideals of $\Gamma$-semigroup and fuzzy soft $\Gamma$-left quasi regular and also to obtain some interesting properties of them.

**Keywords: ** Soft set, fuzzy set ,soft prime, fuzzy soft $\Gamma$-left quasi regular.

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Authors |
: | Pablo I. Pucheta |

Title |
: | On Extended Tempered Fractional Calculus |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 19−23 |

**Abstract: ** In this paper we will introduce an extension of the Riemann-Liouville tempered fractional derivative (integral) replacing the exponential factor by the one parameter Mittag-Leffler function. We will calculate the Fourier Transform of this new tempered fractional operator.

**Keywords: ** Fractional Calculus, Tempered Fractional Calculus, Mittag-Leffler function, Fourier Transform.

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Authors |
: | S. Pious Missier and K. M. Arifmohammed |

Title |
: | Certain Investigations on Digital Plane |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 25−32 |

**Abstract: ** We introduce the concept of $^\#g\hat{\alpha}$-closed sets in a topological space and characterize it using $^*g\alpha o$-kernel and $\tau^\alpha$-closure. Moreover, we investigate the properties of $^\#g\hat{\alpha}$-closed sets in digital plane. The family of all $^\#g\hat{\alpha}$-open sets of $(\mathbb{Z}^2, \kappa^2)$, forms an alternative topology of $\mathbb{Z}^2$. We prove that this plane $(\mathbb{Z}^2, ^\#g\hat{\alpha}O)$ is $T_{1/2}$ and $T_{3/4}$. It is well known that the digital plane $(\mathbb{Z}^2, \kappa^2)$ is not $T_{1/2}$, even if $(\mathbb{Z}, \kappa)$ is $T_{1/2}$.

**Keywords: ** Preopen sets, generalized closed sets, $\alpha$-open sets, $^*g\alpha$-closed sets, $^\#g\hat{\alpha}$-open sets, $T_{1/2}$-spaces, digital lines and digital planes.

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Authors |
: | S. J. Ghevariya |

Title |
: | BSM European Put Option Pricing Formula for ML-Payoff Function with Mellin Transform |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 33−36 |

**Abstract: ** This paper contributes to the valuation of BSM European put option pricing formula for Modified-Log (ML) payoff function using Mellin transform. Panini and Srivastav introduced valuation of option through Mellin transform. Mellin transform is related with extended form of well known Laplace and Fourier transforms.

**Keywords: ** European put option, BSM formula, ML-payoff function, Mellin transform.

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Authors |
: | M. Bhuvaneshwari |

Title |
: | Stochastic Inventory System with Multi Optional Service and Finite Source |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 37−44 |

**Abstract: ** In this article, we consider a continuous review perishable inventory system with a finite number of homogeneous sources of demands. The maximum storage capacity is fixed as $S$. The operating policy is $(s, S)$ policy, that is, whenever the inventory level drops to $s$, an order for $Q(= S- s)$ items is placed. The ordered items are received after a random time which is distributed as exponential. The life time of each items is assumed to be exponential. All arriving customers demand first the essential service (regular service) and some of them may further demand one of other optional services: Type $1$, Type $2$, $\ldots$, and Type $N$ service. The
service times of the essential service and of the Type $j$ $(j = 1, 2, \ldots, N)$ service are assumed to be exponentially distributed. The joint probability distribution of the number of customers in the waiting hall and the inventory level is obtained for the steady state case. We have derived the Laplace-Stieljes transforms of waiting time distribution of customers in the waiting hall. Some important system performance measures in the steady state are derived, and the long-run total expected cost rate is also derived.

**Keywords: ** $(s, S)$ policy, Continuous review, Perishable commodity, Optional service, Markov process, Finite population.

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Authors |
: | S. Usharani |

Title |
: | Supra Continuous $\breve{g}$ in Topological Spaces |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 45−50 |

**Abstract: ** The aim of this paper is to introduce and investigate a new class of sets and functions between topological space called supra $\breve{g}$-open sets and supra $\breve{g}$-closed functions respectively, furthermore introduced the concept of supra $\breve{g}$-open mapping and supra $\breve{g}$-closed maps and investigated several properties of them. Additionally, we relate and compare these functions with some other functions in topological spaces.

**Keywords: ** Supra $\breve{g}$-open set, supra $\breve{g}$-continuity, supra $\breve{g}$-open map, supra $\breve{g}$-closed map and supra topological spaces.

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Authors |
: | N. Srikantha and Venkatesha |

Title |
: | On Geometry of Pseudo-Slant Submanifolds of Conformal Sasakian Manifolds |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 51−59 |

**Abstract: ** In this paper, we study the geometry of pseudo-slant submanifolds of conformal Sasakian manifolds. We give some results on submanifolds of conformal Sasakian manifolds with parallel canonical structures. Finally we discuss integrability conditions of anti-invariant and slant distributions of pseudo-slant submanifolds of conformal Sasakian manifolds.

**Keywords: ** Conformal Sasakian manifolds, integrability, parallel, pseudo-slant submanifolds.

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Authors |
: | V. Puneeth and T. V. Joseph |

Title |
: | On k-Near Perfect Numbers |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 61−65 |

**Abstract: ** A positive integer $n$ is said to be k-near perfect number, if $$\sigma(n)=2n+\sum_{i=1}^{k} d_i$$ where $d_i$'s are proper divisors of $n$ and function $\sigma(n)$ is the sum of all positive divisors of $n$. In this paper we discuss some results concerning with k-near perfect numbers. Near perfect numbers are nothing but 1-Near Perfect Numbers.

**Keywords: ** Divisor Function, Mersenne Prime, Fermat Prime, Perfect Number, Near Perfect Number and k-Near Perfect Number.

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Authors |
: | M. Raja, N. Seenivasagan and O. Ravi |

Title |
: | New Type of Generalized Closed Sets in Topological Spaces |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 67−74 |

**Abstract: ** In this paper two new classes of sets, namely $B\delta g$-closed sets and $B\delta g$-open sets are to be introduced and studied in topological spaces. We prove that this class of $B\delta g$-closed sets lies between the class of $\delta$-closed sets and the class of $\delta g$-closed sets. Also we find some relations between $B\delta g$-closed sets and already existing closed sets. Further we discuss their characterisations and obtain their applications.

**Keywords: ** $B$-set, generalized closed set, $\delta g$-closed set, $\delta$-closure of a subset.

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Authors |
: | N. V. C. Shukla and Ram Asray Verma |

Title |
: | CR-Submanifolds of P-Sasakian manifold Endowed with a Semi-Symmetric Non-metric Connection |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 75−79 |

**Abstract: ** We define a semi-symmetric non-metric connection in a para-Sasakian manifold
and study CR-submanifolds of a para-Sasakian manifold endowed with a
semi-symmetric non-metric connection. Moreover, we also obtain integrability
conditions of the distributions on CR-submanifolds.Parallel horizontal
distributions of CR$-$submanifolds.

**Keywords: ** p-sasakian manifold, CR-Submanifolds, a Semi Symmetric Non metric
Connection, parallel horizontal distribution.

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Authors |
: | P. Sekar and K. Vidhyalakshmi |

Title |
: | Some Decompositions of Weaker Form of Continuity |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 81−89 |

**Abstract: ** In this paper we introduce the notions of
$\alpha$gs-$\mathcal{I}$-open sets, pgs-$\mathcal{I}$-open sets,
sg-$\mathcal{I}$-open sets, $\omega$$_t$-$\mathcal{I}$-sets,
$\omega$$_{\alpha^*}$-$\mathcal{I}$-sets and
$\omega$$_S$-$\mathcal{I}$-sets in ideal topological spaces and
investigate some of their properties. Using these notions we
obtain decompositions of $\omega$-continuity.

**Keywords: ** $\omega$-open set, $\mathcal{I}$-open set, $\omega$-continuity set, $\alpha$g-$\mathcal{I}$-open
set.

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Authors |
: | A. Gnanam and S. Krithika |

Title |
: | Solution of $x^{2} -nx-1=0$ by Continued Fraction Method and Comparison of the Solution by Newton Raphson Method |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 91−96 |

**Abstract: ** In this paper we find the solution of the quadratic equation $x^{2} -nx-1 =0$ using Continued Fraction method and identify the convergence of the solutions with other numerical methods such as Bisection method, False Position method, Iteration method and Newton Raphson method.

**Keywords: ** Continued fractions, Simple continued fraction, Euclidean algorithm, Quadratic irrational, Golden mean, Silver mean, Bisection method, False Position method, Iteration method and Newton -Raphson method.

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Authors |
: | N. Swathi |

Title |
: | Reliability Analysis of Time-Dependent System when the Number of Cycles Follow Geometric Distribution and Stress-Strength Follow Exponential Distribution |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 97−108 |

**Abstract: ** Failure of a system may occur due to certain type of stresses acting on them. If these stresses do not exceed a certain threshold value the system may work for a long period. On the other hand, if the stresses exceed the threshold they may fail within no time. There is uncertainty about stress and strength random variables at any instant of time and also about the behavior of the variables with respect to time and cycles. Time dependent stress- strength models are considered with repeated application of stress and also the change of the strength with time. Reliability of time dependent stress- strength system is carried out by considering each of stress and strength variables are random-fixed. In this paper to find the reliability, components are assumed to be identical and the number of cycles for any time period t is assumed to be random. Expression for system reliability have been attained when number of cycles can be follow Geometric distribution and stress and strength both follow exponential distribution \& computations were also done.

**Keywords: ** Stress-strength system, reliability, Geometric distribution, Exponential distribution, Time dependent.

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Authors |
: | Nistala V.E.S. Murthy and Lokavarapu Sujatha |

Title |
: | Products and Factors of Bresets |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 109−129 |

**Abstract: ** The aim of this paper is a. to generalize the conjunctive and co-normal products of digraphs to an arbitrary family of digraphs or sets with one binary relation each b. to introduce sectional product, for an arbitrary family of digraphs and study their properties and c. to introduce and study properties of 1. conjunctive factor 2. disjunctive factor and 3. sectional factor, for a set with a binary relation whose underlying set is a Cartesian product of a family of sets, thus introducing and elementarily studying factorization theory of digraphs. Further, several relations between these products and factors are studied along with algorithms to compute both products and factors.

**Keywords: ** Digraph, Algorithm, Conjunctive product, Disjunctive product, Sectional product, Conjunctive factor, Disjunctive factor, Sectional factor.

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Authors |
: | R. Raja |

Title |
: | A Criterion for the Secondary Unitary Congruence of Conjugate Secondary Normal Matrices |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 131−134 |

**Abstract: ** In this paper, conjugate secondary normal (con-s-normal) matrices play the same role in the theory of secondary unitary (s-unitary) congruences as conventional s-normal matrices do in the theory of s-unitary similarities. The aim of this section is to propose a simple criterion for s-unitary congruence for the class of con-s-normal matrices.

**Keywords: ** s-normal matrix, con-s-normal matrix, s-unitary, s-eigenvalues.

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Authors |
: | R. Kalaivani and D. Vijayalakshmi |

Title |
: | Dominator Chromatic Number of Derived Graph of Some Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 135−139 |

**Abstract: ** In a graph $G$, the distance $d(u, v)$ between a pair of vertices $u$ and $v$ is the length of a shortest path joining them. The Derived graph, $G^{\dag} $ is the graph whose vertices are same as the vertices of $G$ and two vertices in $G^{\dag} $ are adjacent if and only if the distance between them in $G$ is two. In this paper we obtain the exact value for $\chi_d$ for Derived Graph of Path, Cycle, Sunlet graph, Bistar graph and Triple Star graph respectively.

**Keywords: ** Coloring, Domination, Dominator Coloring, Derived Graph, Double graph.

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Authors |
: | B. Basavanagoud and Praveen Jakkannavar |

Title |
: | Computing First Leap Zagreb Index of Some Nano Structures |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 141−150 |

**Abstract: ** Topological indices have several chemical applications in the quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) analysis of chemical compounds. In this paper, we study the linear regression analysis of the first leap Zagreb index with respect to entropy, acentric factor, enthalpy of vaporization (HVAP), standard enthalpy of vaporization (DHVAP) and boiling point of octane isomers. Finally, we present the expressions for first leap Zagreb index of certain nano structures.

**Keywords: ** First Zagreb index, leap Zagreb index, line graph, subdivision graph, line graph.

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Authors |
: | P. Jeyalakshmi |

Title |
: | Nano $\mathcal{O}_{16}$-closed Sets |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 151−156 |

**Abstract: ** In this paper, we investigated a new class of sets called nano $\mathcal{O}_{16}$-closed sets and nano $\mathcal{O}_{16}$-open sets in nano topological spaces and its properties are studied.

**Keywords: ** Nano $\mathcal{O}_{16}$-closed set, nano $g$-closed set, nano $g \alpha$-closed
sets and nano $gpr$-closed set.

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Authors |
: | S. Bhaumik and S. K. Tiwari |

Title |
: | General Weak Contraction and Some Fixed Point Results on Cone Random Metric Space |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 157−166 |

**Abstract: ** We define weak contraction and general weak contraction on cone random metric space. Then we prove some fixed point results of weak contraction on cone random metric space. The results are verified with the help of suitable example.

**Keywords: ** Weak contraction, General weak contraction, Cone metric space, Cone random metric space.

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Authors |
: | R. Jayagopal and V. Raju |

Title |
: | Domination Parameters in Shadow Graph and Path Connected Graph |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 167−172 |

**Abstract: ** A locating-dominating set (LDS) $S$ of a graph $G$ is a dominating set $S$ of $G$ such that for every two vertices $u$ and $v$ in $V(G) \setminus S$, $N(u)\cap S \neq N(v)\cap S$. The locating-domination number $\gamma^{L}(G)$ is the minimum cardinality of a LDS of $G$. Further, if $S$ is a total dominating set then $S$ is called a locating-total dominating set (LTDS). In this paper, we determine the locating-domination and locating-total domination numbers of the shadow graph and for a special class of path connected graph.

**Keywords: ** Dominating set, total dominating set, locating-dominating set, locating-total dominating set, shadow graph, path connected graph./p>

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Authors |
: | I. Baidari, P. Savant and A. Yalnaik |

Title |
: | On Adriatic Indices and its Application to Some Properties of Alkanes |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 173−181 |

**Abstract: ** The adriatic indices like max-min and min-max deg indices are defined by V.Damir and G.Marija. In this paper we have given certain generalization for some graphs and presented few bounds for these two indices. Along with that some graph operations has been performed like join and corona product. Lastly some chemical applications of max-min and min-max degree indices for alkanes are explored.

**Keywords: ** Adriatic indices, Heat of Atomization, Heat of Formation, Chemical trees.

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Authors |
: | P. Shiladhar, A. M. Naji and N. D. Soner |

Title |
: | Computation of Leap Zagreb Indices of Some Windmill Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 183−191 |

**Abstract: ** Recently, A. M. Naji \cite{Naji1}, introduced leap Zagreb indices of a graph based on the second degrees of vertices (number of their second neighbours). The first leap Zagreb index $L M_1(G)$ is equal to the sum of squares of the second degrees of the vertices, the second leap Zagreb index $L M_2(G)$ is equal to the sum of the products of the second degrees of pairs of adjacent vertices of G and the third leap Zagreb index $L M_3(G)$ is equal to the sum of the products of the first degrees with the second degrees of the vertices. In this paper, we computing Leap Zagreb indices of windmill graphs such as French windmill graph $F^m_n$, Dutch windmill graph $D^m_n$, Kulli cycle windmill graph $C^m_{n+1}$, and Kulli path windmill graph $P^m_{n+1}$.

**Keywords: ** Degrees, Second degrees (of vertex), Leap Zagrab indices, windmill graphs.

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Authors |
: | B. Basavanagoud and E. Chitra |

Title |
: | Degree Square Sum Energy of Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 193−205 |

**Abstract: ** Degree square sum matrix $DSS(G)$ of a graph $G$ is a square matrix of order equal to the order of $G$ with its $(i,j)^{th}$ entry as ${{d_i}^{2}+{d_j}^{2}}$ if $i\neq j$ and zero otherwise, where $d_i$ is the degree of the $i^{th}$ vertex of $G$. In this paper, we obtain the characteristic polynomial of the degree square sum matrix of graphs obtained by some graph operations. In addition, bounds for largest degree square sum eigenvalue and for degree square sum energy of graphs are obtained.

**Keywords: ** Degree square sum matrix, degree square sum polynomial, Eigenvalues, Graph operations.

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Authors |
: | K. Selvakumari and S. Santhi |

Title |
: | A Pentagonal Fuzzy Number in Solving Fuzzy Sequencing Problem |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 207−211 |

**Abstract: ** In this paper, we proposed a solution procedure to solve fuzzy job sequencing problem, where processing time being taken as pentagonal fuzzy numbers. The optimal solution of the job and idle time for each machine is obtained by solving corresponding crisp sequencing problems. The procedure adopted was the fuzzy sequencing problems that are defuzzified using Pascal's triangular graded mean for pentagonal fuzzy ranking index method. The fuzzy sequencing problem has been transformed in to crisp one, using linguistic variable and solved by Johnson's Algorithm. The linguistic variable helps to convert qualitative data in to quantitative data which will be effective in dealing with fuzzy sequencing problems at qualitative nature.

**Keywords: ** Fuzzy sequencing problem, pentagonal fuzzy number, Pascal's Triangular graded mean technique.

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Authors |
: | S. Mehar Banu, S. Nalini and R. Arul |

Title |
: | Oscillation of Third Order Nonlinear Difference Equations With Several Neutral Terms |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 213−221 |

**Abstract: ** This paper deals with the oscillatory properties of third order delay difference equation with several neutral terms. Some new sufficient conditions are derived which are very useful to study the given equation. Examples are provided to illustrate the main results.

**Keywords: ** Third order difference equation, several neutral terms, oscillation.

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Authors |
: | L. Jethruth Emelda Mary and T. Monica |

Title |
: | Domination Number of A Line Graph Formed from the Cartesian Product of Cycle and Path Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 223−232 |

**Abstract: ** In this paper we discussed the domination of line graph formed from the Cartesian product of cycle and path graphs. Also we have introduced disconnected total irregular dominating set and disconnected inverse irregular dominating set and various parameters of domination are applied on line graph and some relation between them are discussed with some examples.

**Keywords: ** Domination number, Inverse Domination number, Total Domination number, disconnected inverse irregular dominating set, disconnected total irregular dominating set Line graph, Cartesian product, Path graph, Cycle graph.

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Authors |
: | Piyush Gupta and Vishal Vincent Henry |

Title |
: | A Study on the Optimization of Transportation Problem |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 233−236 |

**Abstract: ** Determining efficient solutions for large scale transportation problems is an important task in operational research. In this study, Least Cost Method (LCM) which is well known transportation methods in the literature was investigated to obtain more efficient initial solutions.

**Keywords: ** Transportation problem, Least Cost Method, Odd cost, Even cost, Initial Basic feasible solution.

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Authors |
: | K. Thirugnanasambandam and G. Chitra |

Title |
: | Some Results on Modified Mean Labeling of Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 237−240 |

**Abstract: ** Let G be a graph with p vertices and q edges. Let $f : V (G)\to\{1,2,3,\dots,p\}$ be an bijective function. For a vertex labeling f, the induced edge labeling $f^{*}(e = uv)$ is defined by $f^{*}(e) = \frac{f\left(u\right)+f(v)}{2}$ if $f(u) + f(v)$ is even and $\frac{f\left(u\right)+f\left(v\right)-1}{2}$ if $f(u) + f(v)$ is odd, then f is called a modified \textit{mean} labeling if $\{f^{*}(e) / e \in E(G)\} = \{1,2,3,\dots,p-1\}$ and all are distinct integers. In the present work we investigate modified mean labeling of Paths, Caterpillar and Spider.

**Keywords: ** Modified mean Labeling, Odd and Even modified mean labeling, Path, Caterpillar, Spider.

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Authors |
: | M. Ganapathi, S. V. Varma, K. R. K. Prasad and V. B. Kumar |

Title |
: | Stochastic Non-Newtonian Lubrication in Roller Bearings Considering Viscosity Variation and Thermal Effects |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 241−252 |

**Abstract: ** In this paper the generalized Reynolds equation derived for power law fluid is applied to study the squeeze film lubrication of parallel plates considering surface roughness, viscosity variation and thermal effects. The consistency of the lubrication is assumed to vary across the film as two layer fluid and thermal effects by introducing a thermal parameter â€˜qâ€™ in the consistency. Stochastic theory is applied to study roughness effect. Two types of roughness transversal roughness and longitudinal roughness are recognized and expected values of load capacity and time of squeezing are obtained. For all flow behavior index â€˜nâ€™ the effects of high consistency near periphery on these parameters are also studied for both types of roughness.

**Keywords: ** Power law lubrication, consistency, viscosity, thermal effects, load capacity, squeezing time, film thickness etc.

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Authors |
: | Haile Habenom, Hagos Tadesse and G. Venkat Reddy |

Title |
: | Unified Finite Integral Associated with Generalized I-function |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 253−259 |

**Abstract: ** In this paper, the authors established new results by applying definite integrals on the product of $\bar{I}$-function with the hypergeometric function. Several other new and known results can also be obtained from our main theorems.

**Keywords: ** $\bar{I}$-function, hypergeometric function, Definite integrals.

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Authors |
: | D. Sivakumar, C. Loganathan and M. Prakash |

Title |
: | Dynamical Analysis of a Fractional Order Delayed Prey-Predator Model With Stage Structure |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 261−271 |

**Abstract: ** This paper is mainly connected with the investigation of fractional order stage-structured predator-prey system with time delay. By analyzing the corresponding characteristic equations, the local stability of the equilibria is investigated and conditions at which the existence of Hopf bifurcation are derived at positive equilibrium by employing Routh Hurwitz criterion. Both fractional order and time delay are chosen as bifurcation parameters. Further, it is concluded that, if the values of fractional order and time delay exceeds the derived critical value then the solutions of addressed system exhibits oscillatory behavior. Moreover, Lyapunov global stability, complex dynamics of the predator-prey systems are also investigated with or without delay for the incommensurate fractional order. Finally, numerical illustrations are provided to validate the effectiveness of derived analytical results.

**Keywords: ** Prey-predator, Fractional order, Discrete delay, Bifurcation, Global stability.

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Authors |
: | Berhanu Assaye, Mihret Alamneh and Teferi Getachew |

Title |
: | $\delta$-fuzzy Ideals in MS-algebras |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 273−280 |

**Abstract: ** In this paper, we introduce the concept of $ \delta $-fuzzy ideals and $ \alpha $-level principal $\delta $-fuzzy ideals in an MS-algebra and study basic properties of these ideals with respect to homomorphisms. It is observed that the class of all $ \delta $-fuzzy ideals forms a complete distributive lattice and the class of all $ \alpha $-level principal $ \delta $-fuzzy ideals forms a de Morgan algebra. A characterization of $ \delta $-fuzzy ideals in terms of $ \alpha $-level principal $ \delta $-fuzzy ideals is given.

**Keywords: ** MS-algebra, de Morgan algebra, $ \delta $-fuzzy ideals, $ \alpha $-level principal $ \delta $-fuzzy ideals, fuzzy filters, homommorphisms.

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Authors |
: | P. L. Powar, G. R. K. Sahu and Akhilesh Pathak |

Title |
: | Serendipity Fixed Point Using Dual Quasi Contraction |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 281−286 |

**Abstract: ** The concept of Serendipity fixed point was initiated by Powar \cite{1}. Further, Powar \cite{2} established the result of existence of Serendipity Fixed Point using dual F-Contraction. Recently, Poom Kuman \cite{3} extended the result of Ciric and established uniqueness of Fixed Point. Considering the lighter concept of completeness, in the present paper, authors have established the existence and uniqueness of Serendipity Fixed Point under dual quasi and dual generalized quasi contraction condition.

**Keywords: ** Serendipity Fixed Point, weak T-Orbittaly complete Space, dual quasi contraction.

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Authors |
: | T. Thangam and K. Bageerathi |

Title |
: | Generalization of Fuzzy Boundary in Fuzzy Bitopological Spaces |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 287−294 |

**Abstract: ** Focus of this paper is to introduce the concept of fuzzy $(\tau_{i}, \tau_{j})$-boundary and fuzzy $\mathfrak{C}$-$(\tau_{i}, \tau_{j})$-boundary in fuzzy bitopological space where $\mathfrak{C}: [0,1] \to [0,1]$ is a complement function. Several examples are given to illustrate the concepts introduced in this paper.

**Keywords: ** Fuzzy complement function $\mathfrak{C}$, fuzzy $\mathfrak{C}$-$(\tau_{i}, \tau_{j})$-boundary in fuzzy bitopological spaces.

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Authors |
: | D. K. Shirley Gloria and G. Albert Asirvatham |

Title |
: | Context-free Array-token Petri Nets |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 295−304 |

**Abstract: ** Any pure two dimensional context - free Language can be
generated by Array - Token Petri Nets. Some of the closure
properties are obtained.

**Keywords: ** Array-token Petri net (ATPN), context-free grammar (CFG),
pure two dimensional context-free grammar (P2DCFG), pure two
dimensional context-free language (P2DCFL).

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Authors |
: | S. Neelkamal, P. H. Veena and V. K. Pravin |

Title |
: | MHD Momentum Transfer Flow of a Non-Newtonian Walters' Liquid Over a Quadratic Stretching Porous Sheet |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 305−314 |

**Abstract: ** MHD boundary layer momentum transfer flow of a non-Newtonian Visco-elastic Walters` liquid B` model in a saturated porous medium over a quadratic porous stretching sheet have been presented in the paper. The boundary conditions of the problem are constituted by the typical choice of quadratic stretching boundary including a linear mass flux in the velocity normal and a quadratic part in velocity parallel to the boundary sheet through Visco-elastic fluid. The effect of various non dimensional parameters on velocity profiles, stream line patterns and skin friction co-efficient are discussed and presented through graphs.

**Keywords: ** Non-newtonian walters' liquid, stretching sheet,porosity, MHD, streamlines.

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Authors |
: | Nitin J. Wange, M. N. Gaikwad and S. P. Pawar |

Title |
: | Solution for One-Dimensional Transient Heat Conduction And Thermoelasticity in a Multilayer Hollow Sphere |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 315−327 |

**Abstract: ** An analytical solution is obtained for the problem of one dimensional transient heat conduction and thermo elasticity in the multilayered hollow sphere. The sphere has multiple layers in the radial direction and each layer is time dependent and spatially without heat sources are considered. To obtain the temperature distribution The eigen value problem is solved by use of separation of variables method. At $t> 0$ homogenous boundary conditions of the first kind are set on the inner radial surface ($i = 1$, $r=r_{0}$) and third kind are set on the outer ($i= n$, $r= r_n$) radial surfaces (Convection).

**Keywords: ** One dimensional transient heat conduction, Thermal Displacement, Thermal stresses, multilayer sphere.

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Authors |
: | J. Naveen And S. Meena |

Title |
: | Prime Labelings of Some Wheel And Crown Related Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 329−346 |

**Abstract: ** A Graph G with n vertices is said to admit prime labeling if its vertices can be labeled with distinct positive integers not exceeding n such that the labels of each pair of adjacent vertices are relatively prime. A graph G which admits prime labeling is called a prime graph. In this paper we investigate the existence of prime labeling of some graphs related to wheel $W_n$ and crown $C^*_n$. We discuss prime labeling in the context of the graph operation namely corona product.

**Keywords: ** Graph Labeling, Prime Labeling, corona product, Prime Graph.

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Authors |
: | A. Sugumaran and K. Rajesh |

Title |
: | Sum Divisor Cordial Labeling On Some Special Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 347−353 |

**Abstract: ** A sum divisor cordial labeling of a graph $G$ with vertex set $V$ is a bijection $f:V(G)\to \left\{1,2,...,|V(G)|\right\}$ such that each edge uv assigned the label $1$ if $2$ divides $f(u)+f(v)$ and $0$ otherwise. Further, the number of edges labeled with $0$ and the number of edges labeled with $1$ differ by at most $1$. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we prove that the plus graph, umbrella graph, path union of odd cycles, kite and complete binary tree are sum divisor cordial graphs.

**Keywords: ** Divisor cordial labeling, Sum divisor cordial labeling.

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Authors |
: | S. V. Prasad, R. K. Verma, A. Srivastava and D. Gupta |

Title |
: | A Study on Multi Server Queuing Model to Optimize the Performance of a Toll Plaza |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 355−359 |

**Abstract: ** In this paper we studied multi server queuing model and analyze the performance of toll plaza which is located at Khalghat, Agra-Bombay road (on NH 3). The results of the analysis showed that average queue length, waiting time of vehicle at toll plaza. In particular, we present the optimal number of toll booths to reduce the queue length and waiting time of vehicles.

**Keywords: ** Toll plaza, queue length, waiting time, Queuing model.

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Authors |
: | D. K. Thakkar and N. J. Savaliya |

Title |
: | About Isolate Inclusive Sets in Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 361−367 |

**Abstract: ** In this paper we further study isolate domination in graphs. In particular we study the effect of removing an edge from the graph on the isolate domination number of the graph. We prove a necessary and sufficient condition under which the isolate domination number increases when an edge is removed from the graph. Further we also prove a necessary and sufficient condition under which the isoinc number increases when an edge is removed from the graph. We also consider the graphs for which isolate dominating number is equal to 1 or 2.

**Keywords: ** Isolate dominating set, minimal isolate dominating set, minimum isolate dominating set, isolate domination number, isolate inclusive set, private neighbourhood.

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Authors |
: | J. Prakash, K. Kultaran, K. Vinod and S. Vipan |

Title |
: | On the Limitations of Complex Growth Rate in Triply Diffusive Convection in Porous Medium: Darcy-Brinkman Model |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 369−375 |

**Abstract: ** The paper mathematically establishes that the complex growth rate $(p_r, p_i)$ of an arbitrary, neutral or unstable oscillatory perturbation of growing amplitude in triply diffusive convection analogous to stern type fluid layer heated from below, must lie inside a semicircle in the right half of the $(p_r, p_i)$-plane whose centre is at the origin and radius equals $\frac{1}{E\sigma}\left(\sqrt{\left|R\right|E\sigma-\frac{27}{4}p^{4}}\right)$, where $R$ is the thermal Rayleigh number, s is the Prandtl number,$E$ is a constant. Further this result is uniformly valid for quite general nature of bounding surfaces.

**Keywords: ** Triply diffusive convection, Porous medium, Darcy-Brinkman model, Rayleigh number.

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Authors |
: | S. Rubanraj and J. Sangeetha |

Title |
: | Solving Fuzzy Fractional Differential Equation with Fuzzy Laplace Transform Involving Trigonometric Function |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 377−384 |

**Abstract: ** This paper deals with fuzzy Laplace transform to obtain the solution of fuzzy fractional differential equation (FFDEs) under Caputo's H-differentiability. In order to solve FFDEs, first we have to understand about fuzzy Laplace transform of Caputo's H-differentiability with fractional order ($0 <\beta<1$). An analytical solution is presented to confirm the capability of proposed method.

**Keywords: ** Fuzzy fractional differential equation, Fuzzy Laplace transforms, trigonometric function, hyper geometric function.

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Authors |
: | Varta Teotia and Sapna Shrimali |

Title |
: | Some Parameters of Non-Zero Zero Divisor Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 385−388 |

**Abstract: ** Let S be a commutative semigroup including zero. The zero divisor graph associated to S, denoted by $\Gamma(Z)$ is the graph whose vertices are the nonzero zero divisors of S and two vertices x and y are adjacent in case their product in the semigroup is zero. There are many known results on the possible structure of such graphs. We study the converse problem. In this paper, we investigate diameter, radius and cut vertex parameters of some non-zero zero divisor graphs.

**Keywords: ** Commutative semigroup, zero-divisor,graphs, diameter, radius, cut-vertex.

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Authors |
: | R. Chiranjeevi, CH. Mahesh, Uma Dixit and S. Madhulatha |

Title |
: | A Mixed Quadrature Rule of Precision 5 |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 389−394 |

**Abstract: ** A mixed quadrature rule of higher precision for approximate evaluation
of real definite integrals has been constructed using an Lobatto rule.
The analytical convergence of the rule has been studied. The relative
efficiencies of the mixed quadrature rule has been shown with the help
of suitable test integrals. The error bound has been determined asymptotically.

**Keywords: ** Lobatto two point rule, anti-Lobatto three point
rule, Fejer three point second rule, mixed quadrature rule.

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Authors |
: | A. Mondal and A. K. Pal |

Title |
: | Modelling of Synthetic Drugs Transmission with Psychological Addicts |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 395−407 |

**Abstract: ** The synthetic drugs that are making the rounds of drug-using markets are some of the most destructive drugs available today, which have given rise to serious social issues in recent years. In this paper, a synthetic drugs transmission model with psychological addicts and Holling Type II functional responses is proposed. The basic reproduction number $R_0$ is obtained by the method of next generation matrix. Then we have discussed the local and global
stabilities analysis of the model using the basic reproduction number $R_0$. By analyzing the sensitivity of parameters, we obtain that controlling psychological addiction is better than drugs treatment. Computer simulation are carried out to validate our analytical findings. The biological implications of analytical and numerical findings are discussed critically.

**Keywords: ** Synthetic drugs, Basic reproduction number, Local and Global stability, Sensitivity analysis.

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Authors |
: | G. Balasubramanian and B. Ambika |

Title |
: | Edge-Odd Graceful for Cartesian Product of a Wheel With n Vertices and a Path with Two Vertices |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 409−415 |

**Abstract: ** Abhyankar \cite{1} investigated direct methods of gracefully labeling graphs. Bahl \cite{2} got gracefulness labeling for few families of spiders in few terms of merging such graphs. Barrientos \cite{3} obtained graceful labelings for chain graphs. Edwards and Howard \cite{4} analyzed a survey of some classes of graceful trees. Kaneria \cite{6} obtained graceful labeling for graphs related to cycle. Kaneria \cite{7} made new graceful graphs by merging stars. Kaneria \cite{8} received graceful labeling by attaching cycle to cycles and cycle with a complete bipartite graph. Mishra and Panigrahi \cite{10} investigated new classes of graceful lobsters obtained from diameter four trees. Ramachandran and Sekar \cite{11} got graceful labelling of super subdivision of ladder. $A (p, q)$ connected graph is edge-odd graceful graph if there exists an injective map $f: E(G) \to\{1, 3, \dots, 2q-1\}$ so that induced map $f_{+}: V(G)\to \{0, 1,2, 3, \dots, (2k-1)\}$ defined by $f_{+}(x) \equiv\sum f(xy)\ (mod\ 2k)$, where the vertex x is incident with other vertex y and $k = \max \{p, q\}$ makes all the edges distinct and odd. In this article, the edge-odd gracefulness of cartesian product of $P_{2}$ and $W_{n}$ is obtained.

**Keywords: ** Graceful graphs, edge-odd graceful labeling, edge-odd graceful graph.

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Authors |
: | R. Shenbagavalli, G. Balasubramanian and A. Solairaju |

Title |
: | Correlation and Ranking Analysis Using Neutrosophic Sets in a Multi-criteria Single-valued Decision Making Problems |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 417−424 |

**Abstract: ** As a generalization of intuitionistic fuzzy sets, neutrosophic sets (NSs) can be better handle the incomplete, indeterminate and inconsistent information, which have attracted the widespread concerns for researchers. In this paper, some new correlation coefficient and ranking method are introduced using neutrosophic fuzzy operators and fuzzy average operator in a single-valued neutrosophic environment. Firstly, the definition and operational laws of single-valued neutrosophic numbers (SVNNs) are introduced. Then, the single-valued neutrosophic average operator and the single-valued ranking techniques in neutrosophic are developed, and few properties of on these operators are also analyzed. Furthermore, a method for solving multi-criteria decision-making (MCDM) problems is explored based on the correlation coefficient and raking technique. Finally, an illustrative example is shown to verify the effectiveness and practicality of the proposed method.

**Keywords: ** Multi-criteria decision-making, a single-valued neutrosophic sets fuzzy correlation coefficient. Ranking method.

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Authors |
: | U. Vijayachandra Kumar and R. Murali |

Title |
: | Domination in Some Image Graphs |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 425−430 |

**Abstract: ** Let $G$ = $(V,E)$ be a simple connected and undirected graph. A subset $D$ of $V$ is called a dominating set of $G$ if every vertex not in $D$ is adjacent to some vertex in $D$. The domination number of $G$ denoted by $\gamma(G)$ is the minimal cardinality taken over all dominating sets of $G$. The shadow graph of $G$, denoted $D_2(G)$ is the graph constructed from $G$ by taking two copies of $G$, say $G$ itself and $G^{'}$ and joining each vertex $u$ in $G$ to the neighbors of the corresponding vertex $u^{'}$ in $G^{'}$. Let $D$ be the set of all distinct pairs of vertices in $G$ and let $D_{s}$ (called the distance set) be a subset of $D$. The distance graph of $G$, denoted by $D(G, D_{s})$ is the graph having the same vertex set as that of $G$ and two vertices $u$ and $v$ are adjacent in $D(G, D_{s})$ whenever $d(u,v)\in D_{s} $. The image graph of a connected graph $G$, denoted by $I_{mg}(G)$ is the graph obtained by joining the vertices of the original graph $G$ to the corresponding vertices of a copy of $G$. In this paper, we determine the domination number of the image graph of path graph and cycle graph. We also determine the domination number of the shadow distance graph of the image graph of path graph and cycle graph with specified distance sets.

**Keywords: ** Dominating set, Minimal edge dominating set, Minimal Domination number, Shadow distance graph, Image graph.

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Authors |
: | Manjit Singh |

Title |
: | Similarity Reductions and Integrable Properties of Lax Pair for Boiti--Leon--Manna--Pempinelli Equation |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 431−441 |

**Abstract: ** Based on symbolic computational system Maple; classical Lie symmetry reductions of Lax pair for Boiti--Leon--Manna--Pempinelli(BLMP) equation are presented. We have obtained three interesting reductions for Lax pair along with one exact solution. Analysis of reductions have shown that, under similar symmetry group compatibility condition of reduced Lax pair is same as of reduced equation. Moreover, one of the reduced equation is analyzed with Bell polynomials for B$\ddot{\rm{a}}$cklund transformations and Lax pairs; several new traveling wave solutions are also constructed.

**Keywords: ** Lie symmetry, Boiti-Leon-Manna-Pempinelli-equation, Bell polynomials, B$\ddot{\rm{a}}$cklund transformations, Lax pairs, Traveling wave solutions.

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Authors |
: | P.Gayathri and S.Sunantha |

Title |
: | Multiplicative Connectivity Indices of $TUC_{4} C_{8} (R)$ Nanotube |

Volume, Issue, Year |
: | 6(2−B)(2018) |

Pages |
: | 443−455 |

**Abstract: ** Chemical graph theory is an important branch of mathematical chemistry which has wide range applications. In chemical graph theory, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. Degree based topological indices have vital role in chemical graph theory. In this paper we compute the degree based topological indices like Multiplicative first and second Zagreb, Multiplicative first and second Hyper-Zagreb, General first and second Multiplicative Zagreb, Multiplicative Sum connectivity, Multiplicative product connectivity, Multiplicative ABC and Multiplicative GA indices of $TUC_{4} C_{8} $ Rhomboidal Nanotubes.

**Keywords: ** Topological index, molecular graphs, degree based topological index, multiplicative indices, $TUC_{4} C_{8} $ nanotubes.