## Issue 1Volume 6 (2018)

ISSUE 1− A ISSUE 1− B ISSUE 1− C ISSUE 1− D ISSUE 1− E ISSUE 1− Special Issue
 Authors : R. Murali, D.I. Asuntha Rani and A. Ponmana Selvan Title : Generalized Hyers-Ulam Stability for Ladder Network and Fibonacci Sequence Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 1−5

Abstract: We enumerate the Hyers-Ulam-Rassias stability of the homogeneous linear difference equations of Ladder network and Fibonacci sequence with initial conditions by using $Z$-Transforms.

Keywords: Hyers-Ulam-Rassias stability, homogeneous, linear difference equation, $Z$-Transforms method.

 Authors : R. Murali and V. Vithya Title : Stability of Quindecic Functional Equation Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 7−13

Abstract: In this paper, we find the general solution of a quindecic functional equation and prove the stability of quindecic functional equation in matrix normed spaces by using the fixed point method.

Keywords: Generalized Hyers-Ulam stability, J. M. Rassias stability, fixed point, quindecic functional equation, matrix normed spaces.

 Authors : R. Murali, D.I. Asuntha Rani and A. Ponmana Selvan Title : Ulam Stability for Particular Difference Equation of Second Order Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 15−20

Abstract: We study the Hyers-Ulam stability of the homogeneous and non-homogeneous linear difference equation of second order with initial conditions by using $Z$-Transforms method.

Keywords: Hyers-Ulam stability, homogeneous and non-homogeneous, linear difference equation, $Z$-Transforms method.

 Authors : R. Murali and A. Antony Raj Title : Stability of Bi-Cubic Functional Equation in Multi-Banach Spaces Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 21−24

Abstract: In this Paper, We obtain the generalized Hyers-Ulam-Rassias Stability for bi-cubic functional equation in Multi-Banach Spaces.

Keywords: Hyers-Ulam stability, Multi-Banach Spaces, Bi-Cubic Functional Equations, Fixed Point Method.

 Authors : S. Prabhavathi and T. Chandrapushpam Title : Role of Matrices in Cryptography Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 25−28

Abstract: Modern Cryptography exists at the intersection of the disciplines of Mathematics, Computer Science, Electrical Engineering and Communication Science. It is heavily based on mathematical theory and Computer Science practice. One discipline that is being applied in Cryptography is Linear Algebra in specific, Matrices. This paper attempts how to derive the role of matrices in Cryptography in day to day life.

Keywords: Matrices, Inverse Matrices, Congruence,Encryption,Decryption, Plaintext, Cipher text.

 Authors : G. B. Antony Xavier, S. John Borg, S. Jaraldpushparaj and C. Sowmiya Title : Discrete Heat Flow of Rod Controlled by Fibonacci Partial Delay Difference Operator Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 29−32

Abstract: We introduce a new type of Fibonacci discrete heat equation model for rod using Fibonacci difference operator with delay factor $\sigma$. The dispersion of heat is analyzed by Fourier's cooling law and the findings are validated by MATLAB.

Keywords: Partial Difference Equation, Fibonacci Difference Operator, Fibonacci Numbers and Discrete Heat Flow.

 Authors : G. Britto Antony Xavier, M. Meganathan and E. L. Gohila Title : Finite Fourier Series Decomposition of Functions (Signals) of Three Variables Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 33−39

Abstract: In this paper, we apply the properties of discrete orthonormal system of functions and derive certain results on trigonometric functions by using inverse of generalized difference operator. Also we develop a method to find out finite Fourier series decomposition of real valued functions of three variables

Keywords: Discrete orthonormal system, Finite Fourier series, Generalized difference operator and signal processing.

 Authors : G. Britto Antony Xavier, S. John Borg and S. Jaraldpushparaj Title : Discrete Heat Flow Controlled by Alpha-Beta Delay Q-Difference Equations Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 41−46

Abstract: Partial q-difference equation have extended applications in q-heat equations. Having introduced the Fibonacci difference equation in the partial section by using Fibonacci difference operator with shift values, a model for heat tranfer in the rod is found having recourse to Fourier law of cooling, an involved study is carried out to evaluate that movement of heat and thus numerous are postulated. The result obtained are validated by MATLAB.

Keywords: Partial Difference Equation, Partial Difference Operator and Delay Heat Equation.

 Authors : S. Sekar and G. Mayelvaganan Title : Stability of a Quadratic Functional Equation in IFNS Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 47−53

Abstract: In this paper, we study the generalized Hyers-Ulam-Rassias stability for the Quadratic type functional equation in IFNS. $f(x+y)-f(x-y)=f(2x+y)-4f(x)-f(y)$.

Keywords: Quadratic functional equation, Intuitionistic fuzzy normed spaces,Hyers-Ulam stability.

 Authors : R. Murali and V. Vithya Title : Generalized Hyers-Ulam Stability of an Additive Functional Equation Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 55−59

Abstract: In this paper is to obtain some results for the stability of new additive functional equation in matrix normed spaces with the help of fixed point method.

Keywords: Generalized Hyers-Ulam stability, J. M. Rassias stability, fixed point, additive functional equation, matrix normed spaces.

 Authors : R. Murali and A. Ponmana Selvan Title : Ulam Stability of a LCR Electric Circuit with Electromotive Force Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 61−66

Abstract: We enumerate the approximate solution of the second order differential equation of the LCR electric circuit. That is, we study the Hyers-Ulam stability and Hyers-Ulam-Rassias of the dirrential equations $l''(t) + \dfrac{R}{L} \ l'(t) + \dfrac{1}{LC} \ l(t) = 0$ and the non-homogeneous differential equation $l''(t) + \dfrac{R}{L} \ l'(t) + \dfrac{1}{LC} \ l(t) = p(t)$, with initial conditions $H(a) = H'(a) = 0$, where $R, L, C$ are constants and $l \in C^2 (I)$, $p(t) \in C(I)$, $I = [a, b] \subseteq \mathbb{R}$.

Keywords: Hyers-Ulam Stability, Hyers-Ulam-Rassias stability, linear differential equations, homogeneous and non-homogeneous.

 Authors : Javeria Amreen Title : On Signless Laplacian Energy and its Variations Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 67−74

Abstract: The signless Laplacian energy of a simple connected graph $G$ of order n and size m is defined in \cite{ivan2009} as $LE^{+}(G) = \sum\limits_{i=1}^{n} \mid {\lambda_{i} - \frac{2m}{n}} \mid$, where $\lambda_{i}, i=1,2,\dots ,n$ are the eigen values of the signless Laplacian matrix $L^{+}(G)$. In this paper, a new concept called signless Laplacian maximum eccentricity energy is introduced and value of the same is determined for some families of graphs. Also, signless Laplacian energy and signless Laplacian maximum eccentricity energy are compared for some classes of graphs. Further, signless Laplacian energy of direct product of graphs is obtained.

Keywords: Graph energy, signless Laplacian energy, signless Laplacian maximum eccentricity energy.

 Authors : J. Jhonson Savarimuthu and P. Abirami Title : Octagonal Fuzzy Number and its Corresponding Matrices Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 75−82

Abstract: In this paper we defined Octagonal Fuzzy Number(OFN) in continuation with the other defined fuzzy numbers. We also include basic arithmetic operations of octagonal fuzzy numbers. Finally we defined Octagonal Fuzzy Matrix(OFM) with some matrix properties.

Keywords: Fuzzy Numbers, Octagonal Fuzzy Numbers, Octagonal Fuzzy Matrices, Octagonal Fuzzy Determinant.

 Authors : M. Chitra and D. Karthikeyan Title : Mathematical Modeling on Unsteady MHD Oscillatory Blood Flow in an Inclined Tapered Stenosed Artery with Permeable Wall: Effects of Slip Velocity Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 83−89

Abstract: The purpose of this work is to study the Mathemetcal modeling on unsteady MHD Oscillatory blood flow in an inclined tapered stenosed artery with permeable wall the effets of slip velocity. The governing equation are solved by analytically by using Bessel function. The analytic explicit expressions of axial velocity, volumetric flow rate and wall shear stress are given. The effects of slip condition, magnetic field, permeable wall and Oscillatory flow have been discussed. The obtained results for different values of parameters in to the problem under consideration, show that the flow is appreciably influenced by the presence of Reynolds number $(R_e)$, Slip parameter $(\gamma)$, Froude number $(F_r)$, artery inclination $(\alpha)$, inclination of Magnetic field $(\theta)$ , Darcy number $(D_a)$, permeability $(k)$ and Hartmann number $(M)$. Some of the found results show that the flow patterns with slip effects in non-tapered region $(\xi=0)$, converging region $(\xi<0)$ and diverging region $(\xi>0)$ are effectively influenced by the presence of magnetic field and change in learning of artery, with time varying pressure gradient. Results for the effect of permeability on these flow characteristics are shown graphically and discussed briefly.

Keywords: Darcy number, Bessel function, permeable wall , Inclined tapered artery, Slip velocity.

 Authors : P. Jagadeeswari, K. Ramanathan and K. Manimekalai Title : Square Difference Labeling for Pyramid Graph and its Related Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 91−96

Abstract: In this paper, we prove the characterization of pyramid graph $(n \geq 3)$, hanging pyramid graph and the associated graphs admits square difference labeling.

Keywords: Square difference labeling, Pyramid graph, Hanging pyramid, Path union, One point union.

 Authors : G. Subashini, K. Ramanathan and K. Manimekalai Title : Square Sum Labeling for Pyramid Graph and Hanging Pyramid Graph Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 97−102

Abstract: In this paper we prove that square sum labeling for $Py_n$; $(n \geq 3)$, $HPy_n$, $P(r, Py_n)$, $P(r, HPy_n)$, one point union of $r$ copies of pyramid and hanging pyramid graphs are square sum graphs by using Breadth First Search Algorithm.

Keywords: Square sum labeling, Pyramid graph, Hanging Pyramid, Path union, One point union, BFS Algorithm.

 Authors : J. Arthy, K. Ramanathan and K. Manimekalai Title : Difference Cordial Labeling of Arrow Graph Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 103−108

Abstract: In this work we establish the difference cordial labeling of arrow graphs $A_n^2$, $DA_n^2$, $A_n^m$ and $DA_n^m$.

Keywords: Difference cordial graph, arrow and double arrow graphs.

 Authors : C. T. Nagaraj, C. Y. Ponnappan and G. Prabakaran Title : Multiple Magic Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 109−115

Abstract: Let $G$ be a graph of order $n$ and size $m$. A vertex magic total labeling is an assignment of the integers $1, 2, 3, ... ,m+n$ to the vertices and the edges of $G$, so that at each vertex , the vertex label and the labels on the edges incident at that vertex, add to a fixed constant called the magic constant of $G$. Such a labeling is a-vertex multiple magic if the set of the labels of the vertices is $\{a, 2a, ..., na\}$ and is b-edge multiple magic if the set of labels of the edges is $\{b, 2b, ..., mb\}$. This article, presents properties of $a-$ vertex multiple magic graphs and $b-$ edge multiple magic graphs.

Keywords: $a-$vertex multiple magic graphs, $b-$edge multiple magic graphs.

 Authors : J. Jhonson Savarimuthu and N. Karthika Title : Hexagonal Fuzzy Numbers and its Corresponding Matrices Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 117−124

Abstract: In this paper, we define Hexagonal fuzzy number (HFN) in continuation with the other defined fuzzy numbers. We also include basic arithmetic operations of Hexagonal fuzzy numbers. Finally we define Hexagonal fuzzy matrix with some matrix properties.

Keywords: Fuzzy Numbers, Hexagonal Fuzzy Numbers, Hexagonal Fuzzy Matrices.

 Authors : M. Antony Arockiasamy and V. Pavithra Title : Product Cordial Labeling of n-chain Aztec Diamond Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 125−129

Abstract: A function $f: V$ to $\left\{0, 1\right\}$ of a graph G is known to be a product cordial labeling if each edge $uv$ is given the label $f (u) f (v)$ the resulting number of vertices with labels 0 and the number of vertices with labels $1$ vary to the maximum of $1$, and the number of edge labels with $0$ and the number of edge labels with $1$ vary also to the maximum of $1$. A graph that satisfies the conditions of product cordial labeling is named product cordial. In this paper product cordial labeling is proved for $n$-chain Aztec diamond graph for even positive integer $n$.

Keywords: Product cordial labeling, Aztec diamond, Aztec diamond graph, n-chain Aztec diamond graph.

 Authors : Ann Kurillose and Mayamma Joseph Title : Holes in L(3, 2, 1)-Labeling Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 131−137

Abstract: An $L(3, 2, 1)$-labeling is a simplified model for the channel assignment problem. Given a graph G, an $L(3, 2, 1)$-labeling of G is a function $f$ from the vertex set $V(G)$ to the set of all non-negative integers such that $|f(u)- f(v)|\geq 1$ if $d(u,v)=3$, $|f(u)- f(v)|\geq 2$ if $d(u,v)= 2$ and $|f(u)- f(v)|\geq 3$ if $d(u,v)=1.$ The \textit{span} of a labeling $f$, is the difference between the largest label and the smallest label in an $L(3 ,2, 1)$-labeling. The $L(3, 2, 1)$-\textit{labeling number} of G, denoted by $\lambda_{3, 2, 1} (G)$, is the minimum span of all $L(3, 2, 1)$-labelings of G. A \textit{span labeling} is an $L(3, 2, 1)$-labeling whose largest label is $\lambda_{3, 2, 1}(G)$. Let $f$ be an $L(3, 2, 1)$-labeling that uses labels from 0 to $\lambda_{3, 2, 1}(G).$ Then $h \in (0, \lambda_{3, 2, 1}(G) )$ is a \textit{hole} if there is no vertex $v\in V (G)$ such that $f (v) = h$. In this paper, we investigate maximum number of holes in $L(3, 2, 1)$ span labeling of certain classes of graphs.

Keywords: $L(2, 1)$-labeling, $L(3, 2, 1)$-labeling, holes.

 Authors : M. Antony Arockiasamy, S. Kalaiarasi and P. Tamizharasi Title : Cordial Labeling of n-Chain Aztec Diamond Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 139−146

Abstract: A binary vertex labeling $f: V(G)\rightarrow\left\{0,1\right\}$ of a graph G is called a cordial labeling if $|v_f (0)-v_f (1)|\leq1$ and $|e_f (0)-e_f (1)|\leq1$. A graph G is cordial if it admits cordial labeling.. The dual graph obtained from an Aztec diamond of order n, where each square is a vertex and if two squares are adjacent in Aztec diamond then their corresponding vertices are linked by and edge in the dual graph; is known as Aztec diamond graph of order n. In this paper n-chain Aztec diamond graphs are proved to be cordial.

Keywords: Cordial labeling, Aztec diamond, Aztec diamond graph. Disjoint union of graphs

 Authors : G. Uma Maheswari, G. Margaret Joan Jebarani and V. Balaji Title : Sub Super Mean Labeling on a General Three Star Graph Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 147−153

Abstract: In this paper, we consider a three star graph $G={{K}_{1,\,\,\ell }}\cup {{K}_{1,\,\,m}}\cup {{K}_{1,\,\,n}},\,\,\,\ell \le m\le n$ with a $SSML\left( V(4),E(2) \right)$, for $g=\left[ \dfrac{n+\ell -m}{2} \right],$ and find the omissions and repetitions in terms of g for two different cases.

Keywords: Super mean labeling, super mean graph, Sub super mean labeling, FEIO, SEIO, FOIO and star.

 Authors : Shilpa Paul and Mayamma Joseph Title : On Multiplicative Geometric Arithmetic Index of Some Chemical Structures Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 155−163

Abstract: Graph theoretical study of chemical structures is widely used in physico-chemical characterization of molecules. Several numerical quantities known as topological indices that are associated with the graph structures of molecules have been used as a tool for the characterization of chemical structures. Mathematically, a topological index is a function from any structural property of the graph to a positive real number. With many new developments in chemistry, topological indices are widely used in Quantitative Structure Property Relationship (QSPR) studies. For a molecular graph $G(V,E)$ let $d(u) \text{ and } d(v)$ denote the degree of vertices $u,v\in V(G)$ respectively and $uv\in E(G).$ The Geometric Arithmetic index is a degree based topological index defined as, $GA_{1}(G)=\sum\limits_{uv\in E(G)}\frac{\sqrt{d(u)d(v)}}{\frac{d(u)+d(v)}{2}}$. Its multiplicative version, called the Multiplicative Geometric Arithmetic index was introduced recently which is defined as, $GAII(G)=\prod\limits_{uv\in E(G)}\frac{\sqrt{d(u)d(v)}}{\frac{d(u)+d(v)}{2}}$. In this paper, we determine the exact value of \textit{GAII} for some common polycyclic organic structures, namely graphene, triangular benzenoids, benzenoid series, benzenoid systems and phenylenes. Further we obtain a comparison between $GA_{1}$ and $GAII$ indices.

Keywords: Molecular graphs, Topological index, Geometric Arithmetic index, Multiplicative Geometric Arithmetic index.

 Authors : A. Arokiamary and P. Anbarasi Title : Balanced Intuitionistic Triple Layered Fuzzy Graph Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 165−174

Abstract: A new balanced intuitionistic triple layered fuzzy graph is defined using intuitionistic triple layered fuzzy graph by modifying its conditions. Also some properties of BITLFG are verified in this paper.

Keywords: Intuitionistic fuzzy graph, triple layered fuzzy graph, complete fuzzy graph, balanced fuzzy graph, balanced intuitionistic triple layered fuzzy graph.

 Authors : L. Jethruth Emelda Mary and P. Amutha Title : Structural Core Graph of Triple Layered Fuzzy Graph Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 175−184

Abstract: The Triple Layered Fuzzy Graph (TLFG) gives the $3-D$ structure to fuzzy graph. In this paper, we constructed the structural core graph for the given TLFG using a new algorithm and also the structural core graph for the union of two TLFG is also constructed using the same algorithm. Some of its diagrammatic properties are studied.

Keywords: Fuzzy graph, Triple layered fuzzy graph, face value, structural core graph.

 Authors : Anisha Jean Mathias Title : Edge Domination Number on Graph Variations Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 185−193

Abstract: An edge dominating set D is a set of edges from the edge set E in graph G where every edge of $\langle E-D\rangle$ is adjacent to an element of $D$. Edge domination number is the cardinality of the smallest edge dominating set and is denoted by $\gamma ' (G)$. In this study, results for edge domination number of product graphs have been obtained and their algebraic properties have also been examined. Further, we introduce the concept of edge domination subdivision number and determine its value for some graphs and explore the relationship between edge domination subdivision number and edge domination number.

Keywords: Edge Domination number, independent edge domination number, edge domination subdivision number, matching subdivision number.

 Authors : M. Antony Arockiasamy, V. Saraswathi and P. Sathya Title : Cordial And Product Cordial Labeling Of Edge Merged And Vertex Merged n-Chain Aztec Diamond Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 195−202

Abstract: A binary vertex labeling function $f$ from the vertices of a graph $G$ to ${\left\{0,1\right\}}$ is called cordial labeling, if each edge xy is given the label $\left|f(x)-f(y)\right|$, the resulting number of vertices with labels $0$ and the number of vertices with labels $1$ differ to the maximum of $1$, and the number of edge labels with $0$ and the number of edge labels with $1$ also differ to the maximum of $1$. In this paper $n$-chain edge merged and vertex merged Aztec diamond graphs are proved to be cordial but not product cordial.

Keywords: Aztec Diamond Graph, Cordial labeling, Product cordial labeling, Chain of vertex merged Aztec diamond graph, Chain of edge merged Aztec diamond graph.

 Authors : L. Jethruth Emelda Mary and T. Monica Title : Star Colouring of Line Graph Formed From the Cartesian Product of cycle and Path graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 203−212

Abstract: In this paper we determined the star chromatic number of the line graph which is formed from the Cartesian product of path and cycle graphs. Also the relationship between the chromatic number and star chromatic number are analysed and illustrated with some examples.

Keywords: Path graph, Cycle graph, Cartesian product of simple graph, Line graph, Proper colouring, Chromatic number, Starcolouring, Star chromatic number.

 Authors : T.Henson and N. Devi Title : On Direct Sum of Three Fuzzy Graphs Volume, Issue, Year : 6(1)(Special Issue)(2018) Pages : 213−219

Abstract: In this paper, the direct sum $G_1\oplus G_2\oplus G_3$ of three fuzzy graphs $G_1, G_2$ and $G_3$is defined. The degree of the vertices in $G_1\oplus G_2\oplus G_3$ is calculated. The regular property, connectedness and effectiveness on the direct sum of three fuzzy graphs are also discussed.

Keywords: Fuzzy graph, Direct sum, Degree of vertices in the direct sum, Regular fuzzy graph, Connected fuzzy graph, Effective fuzzy graph.