## Issue 1Volume 9 (2021)

 Authors : Shweta Wasnik and Subhashish Biswas Title : Gregus Type Fixed Point Theorems Satisfying Strict Contractive Condition of Integral Type Volume, Issue, Year : 9(1)(2021) Pages : 1−8

Abstract: In the present paper, we prove some common fixed point theorems of Gregus type for two pairs of self mappings satisfying strict contractive condition of integral type by using the weak subsequential continuity property with compatibility of type $(E)$ in metric spaces. Our results improve some previous known results and relevant literature.

Keywords: Gregus type fixed point, weakly subsequentially continuous, compatible of type (E).

 Authors : Vedanti Bhargava Title : Mathematical Model for Predicting Time of a Shared Common Ancestor Volume, Issue, Year : 9(1)(2021) Pages : 9−18

Abstract: Accurate and efficient model development in the field of evolutionary biology is becoming increasingly important to study phylogenetic relationships. Studying these phylogenetic relationships can provide greater insight into areas such as drug development, analyzing disease transmission, and studying the host-pathogen evolutionary relationship. This study uses the idea of a transition matrix to predict how long ago two species shared a common ancestor. Using this concept and various linear algebraic ideas, a mathematical model that simulated amino acid mutations was developed, and it was implemented in Python to perform trials. The main result of the development of the model was a matrix containing the expected values for the number of years ago two amino acids were the same. Simulations of the model can be run in Python multiple times and the average of the values obtained gives an estimate of how many years ago two amino acids were the same. These represent the number of years ago the species shared a common ancestor. The result varies based on the species data analyzed. This model provides a foundation for development of more complex models that enhance phylogenetic tree development. As stated previously, this can have substantial impacts on biotechnology and biomedical sciences, which often rely on analyzing these evolutionary relationships.

Keywords: Transition matrix, Amino Acids, Probability, Mutations, Sequences, Simulations, Expected value.

 Authors : A. M. Nour, M. Manjunatha Title : K-hub Number of a Graph Volume, Issue, Year : 9(1)(2021) Pages : 19−26

Abstract: In this paper, we introduce the concept of $k-$hub set and $k-$hub number of a graph. We compute the $k-$hub number for some standard graphs, also we determined the $k-$hub number for corona of two graphs. Some bounds of $k-$hub number are established. Finally we characterize the structure of all graphs for which $h_k(G)=1$.

Keywords: Hub number, K-hub number, Hub set.

 Authors : Mitesh J. Patel and G. V. Ghodasara Title : On Permutation Labeling of Graphs Volume, Issue, Year : 9(1)(2021) Pages : 27−33

Abstract: An injective function $f:V(G)\rightarrow{}\{1,2,\ldots,|V(G)|\}$ is said to be permutation labeling if each edge $uv$ is assigned with label $\Perm{f(u)}{f(v)}= \ \frac{(f(u))!}{| f(u)-f(v)|! } \ (f(u)>f(v))$ are all distinct. A graph which admits permutation labeling is called permutation graph. In this paper we prove that wheel graph, restricted square and degree splitting graph of bistar graph are permutation graphs. We also proved that arbitrary super subdivision of path graph, star graph and cycle graph are permutation graphs.

Keywords: Permutation labeling, Wheel graph, Bistar graph, Arbitrary super subdivision.

 Authors : Anil Kashyap, Subhashish Biswas and Rajoo Title : To Study Linear with Integer Programming and Solving Multi-Objective Optimization Volume, Issue, Year : 9(1)(2021) Pages : 35−39

Abstract: Here we present a new exact approach for solving Multi-Objective Optimization with. Linear programs and integer programs are optimization problems with linear impartial functions and linear constraints. These problems are actual general: linear programs can be used to express an inclusive range of problems such as two-part corresponding and network stream, while integer programs can express an even larger range of problems, including all of those in model of operation research. We define each of these formalisms in turn.

Keywords: Linear Programming, Integer Programming, multi-objective Optimization, approximation algorithms.

 Authors : F. F. Ngwane Title : Effects of Item Parameter Distributions on Test Information Function Volume, Issue, Year : 9(1)(2021) Pages : 41−50

Abstract: Test information function (TIF) plays an important role in designing a test. The TIF does not only depend on the logistic model such as the 1PL, 2PL and 3PL logistic models, but it also on the parameter distributions of the model. In this study we conducted simulations using the R statistical software on different examinee sizes and different exam length, and we compare the results when using uniform parameter distribution and normal parameter distribution. The findings of this study illustrate the following: First, when using item parameters that are all uniformly distributed the obtained test information function is significantly small compared to the test information function obtained when we use item parameters that all follow a normal distribution. Second, the test information function increases as the length of the exam increases when we use item parameters that are all normally distributed. Third, this study also suggest great flexibility in writing an exam as we can choose the item parameter distribution depending on the goals of the test.

Keywords: Item-Response Theory (IRT), Item Information Function, Test Information Function, one-, two-, and three-parameter models.

 Authors : Puspita Mahata, Sujit Kumar De and Gour Chandra Mahata Title : Optimal Ordering and Screening Policy for Perishable Items with Imperfect Quality Volume, Issue, Year : 9(1)(2021) Pages : 51−61

Abstract: In reality, perishable products also contain defective ones before storage due to uncertainties during production process. In order to reduce the loss caused by the defective products, an inspection process is implemented by retailers. However, the screening is not always perfect due to the limited technical level. In view of the above, a retailer's perishable inventory model with defective items is established in this article. Two types of inspection errors, non-100\% screening and sales returns are all considered in the proposed model. The objective of the model is to minimize the retailer's expected cost per unit time and to find the optimal retail price, the screening rate and the order quantity. Numerical example and sensitivity analysis are provided to illustrate the presented model and provide some managerial insights.

Keywords: Inventory strategy, Deteriorating items, Defective products, Screening.

 Authors : V. Durai Murugan, R. Seethalakshmi and P. Namasivayam Title : A Study on Complementedness in the Subgroup Lattices of $2\times 2$ Matrices Over $Z_{11}$ Volume, Issue, Year : 9(1)(2021) Pages : 63−65

Abstract: In this paper, we verify the complementedness in the subgroup lattices of the group of $2 \times 2$ matrices over $Z_{11}$.

Keywords: Matrix group, subgroups, Lagrange's theorem, Lattice.

 Authors : J. C. Kanani and V. J. Kaneria Title : Cordial Labeling for Eight Sprocket Graph Volume, Issue, Year : 9(1)(2021) Pages : 67−72

Abstract: We obtained a new graph which is called Eight Sprocket graph. We proved that the Eight Sprocket graph is cordial. We have investigated some Eight Sprocket graph related families of connected cordial graphs. We proved that path union of Eight Sprocket graph, cycle of Eight Sprocket graph and star of Eight Sprocket graph are cordial.

Keywords: Cordial labeling, Eight Sprocket graph, Path union of graphs, Cycle of graphs, Star of a graph.

 Authors : Annela R. Kelly Title : Optimal Distribution of Players into Two Alliances in Mis\{e}re Subtraction Game Volume, Issue, Year : 9(1)(2021) Pages : 73−82

Abstract: In combinatorial game theory, multi-player games expands on the research of two-person games and are widely applicable to computer science and business. The given paper studies the mis\{e}re version of the subtraction game (Nim with one pile of counters) for two alliances extending the results of Kelly in \cite{2,3}; Liu and Zhao in \cite{5}, Liu and Wang in \cite{6} and Suetsugu in \cite{7}. The main result of the current paper determines the optimal distribution of the players among two alliances that minimizes $N$, the smallest number of counters such that the larger alliance has a winning strategy for all the games with more than $N$ counters.

Keywords: Multi-player impartial game, Mis\{e}re Subtraction game, Alliances.

 Authors : M. Mohanasundari and K. Mohana Title : Quadripartitioned Neutrosophic Mappings with its Relations and Quadripartitioned Neutrosophic Topology Volume, Issue, Year : 9(1)(2021) Pages : 83−93

Abstract: The purpose of this paper is to construct a quadripartitioned neutrosophic mappings with its relations which is a generalization of single valued neutrosophic mapping. And also studied the properties of quadripartitioned single valued neutrosophic mappings. Further we introduce the concept of quadripartitioned single valued neutrosophic topological spaces and define the basic concepts with examples in detail.

 Authors : V. Mohan Babu and C. Pandurangappa Title : Similarity Solution of Non-Newtonian Three-dimensional Boundary-layer Through Porous Medium Volume, Issue, Year : 9(1)(2021) Pages : 95−104

Abstract: An analysis is made to study a three-dimensional boundary-layer flow of non-newtonian fluid over a porous media. The governing partial differential equations of momentum are transformed into self-similar non-linear ordinary differential equations by using new form of suitable similarity transformations. The obtain equations are then should by using shooting method. The flow phenomena is characterized by the power-law index n', permeability of porous media $\omega$' porosity parameter $\phi$' effective viscosity $\lambda$', three-dimensionality parameters $\alpha$'. The numerical results of the velocity profiles are obtained and displayed graphically.

Keywords: Porous medium, Power-law fluid, Effective viscosity, Porosity, Similar solution.

 Authors : V. Naga Raju Title : Common Fixed Point Theorems for Mappings Satisfying Common Limit Range Property Volume, Issue, Year : 9(1)(2021) Pages : 105−111

Abstract: In this paper, we define generalized weakly contractions in G-metric spaces with which we prove two common fixed point theorems for self maps satisfying common limit range property .The obtained results extend and improve some results in the literature. Further, we deduce some consequences of the results.

Keywords: G-metric space, fixed point, common limit range property, weakly compatible mappings.

 Authors : Meera Meghpara, H. M. Karavadiya and V. J. Kaneria Title : A Study on $\alpha-$graceful Labeling of Spade Graph Volume, Issue, Year : 9(1)(2021) Pages : 113−117

Abstract: In this paper we have derived a new graph named spade graph. We have proved two copies of spade graph joined by an arbitrary path is $\alpha-$graceful, $n$ copies of spade graph joined by any vertex is $\alpha-$graceful, path union of $n$ copies of spade graph is $\alpha-$graceful and open star of finite copies of spade graph is also $\alpha-$graceful.

Keywords: Graceful graph, $\alpha-$graceful graph, path union of a graph $G$, open star of a graph $G$, spade graph.

 Authors : C. K. Ajay and A. H. Srinivasa Title : Temperature Dependent Viscosity Effects on Unsteady MHD Accelerating/Decelerating Flow over a Wedge Volume, Issue, Year : 9(1)(2021) Pages : 119−126

Abstract: The present work brings into focus the unsteady MHD two dimensional accelerating/decelerating boundary layer flows and heat transfer of an incompressible electrically conducting fluid above a moving wedge in the existence of variable viscosity. Suitable transformation is used to form a system of coupled non linear partial differential equations for governing both the flow and heat transfer. These equations have been carried out numerically, which is done by utilizing an implicit finite difference method in combination with quasilinearization technique. The obtained numerical results have been presented graphically in terms of local nusselt number, skin friction, temperature distribution, and velocity distribution for different values of magnetic field (M) and variable viscosity ($\varepsilon$) parameters along with the fixed Prandtl number (Pr) and wedge angle (m). It is found that there exist a unique solution for accelerating flow and dual solutions for decelerating flow.

Keywords: Accelerating flow, Decelerating flow, Heat transfer, Skin friction, Variable Viscosity.