## Volume 4 (2016)Article in Press

ISSUE 2 − A ISSUE 2 − B ISSUE 2 − C ISSUE 2 − D
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 Article Type : Research Article Title : An $M/G/1$ Retrial Queue with Multiple Working Vacation Country : India Authors : S.Pazhani Bala Murugan and K.Santhi

Abstract: We consider an $M/G/1$ retrial queue with general retrial times and multiple working vacation. During the working vacation period, customers can be served at a lower rate. Both service times in a vacation period and in a service period are generally distributed random variables. Using supplementary variable method we obtain the probability generating function for the number of customers and the average number of customers in the orbit. Further more, we carry out the waiting time distribution and some special cases of interest are discussed. Finally, some numerical results are presented.

Keywords: Retrial queues, Working vacation and Supplementary variable method.

S.Pazhani Bala Murugan
Mathematics Section, Faculty of Engineering and Technology, Annamalai University, Annamalainagar, Tamilnadu, India.
E-mail: spbmaths@yahoo.co.in

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 Article Type : Research Article Title : Collatz Conjecture for Modulo an Integer Country : India Authors : T.Kannan and C.Ganesa Moorthy

Abstract: A function $T_m$ from a set \{1,2,3,...$m$\} into itself defined by \\$T_m(x)=\frac{x}{2},$ for even $x$ and by $T_m(x)=\frac{3x+1}{2}$ (mod $m$), for odd $x$ is considered in this article. The asymptotic behaviour of this function is studied in this article for some cases.

Keywords: Congruence, Collatz conjecture.

T.Kannan
Department of Mathematics, Sree Sevugan Annamalai college, Devakottai, TamilNadu, India.
E-mail: hardykannan@gmail.com

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 Article Type : Research Article Title : Inventory Model for Deteriorating Items Involving Trade Credit Policy in Three - Echelon Supply Chain System Country : India Authors : D.Sharmila and R.Uthayakumar

Abstract: In present study, we generalize order linked trade credit policy in three echelon supply chain system where manufacturer, distributor and retailer are involved. Here manufacturer provide delay period to distributors also distributor provide trade credit policy to his retailers. In this paper, we discuss a three echelon supply chain system as cost minimization to determine the system's optimal cycle time. We determine the optimal order time, order quantity and optimal payment time. To investigate the effect of changes in inventory parameter values on the optimal policy, a sensitivity analysis is conducted.

D.Sharmila
Department of Mathematics, The Gandhigram Rural Institute - Deemed University, Gandhigram, Tamilnadu, India.

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 Article Type : Research Article Title : Balanced Mean Cordial Labeling and Graph Operations Country : India Authors : V J Kaneria, M J Khoda and H M Karavadiya

Abstract: Balanced mean cordial labeling is a mean cordial labeling $f$ with \linebreak $|v_{f}(i)- v_{f}(j)|=0$, $|e_{f}(i)- e_{f}(j)|=0$, $\forall$ $i,j\in \{0,1,2\}$. In this paper, we investigate mean cordial labeling for P($t\cdot H$), where H be any graph and $t\equiv 0$(mod $3$). We also investigate balanced mean cordial labeling for TP($t\cdot H$), G$^{*}$, P$_{t}$$\times G, C_{t}$$\times G$, where H and G both are balanced cordial graphs.

Keywords: Path Union, Mean Cordial, Balanced mean cordial.

M J Khoda

E-mail: mjkmaths@gmail.com

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 Article Type : Research Article Title : Nano Regular Generalized Star $b$-Continuous Function in Nano Topological Spaces Country : India Authors : A.Maheswari and M.Sheik John

Abstract: In this paper, we introduce and investigate the notions of nano regular generalized star $b$ continuous functions in terms of nano regular generalized star $b$ closed sets in nano topological spaces.

Keywords: Nano topology,$Nr$-continuous,$Ng^{*}$-continuous,$Nrg^{*}$-continuous,$Nrg^{*}b$-continuous,$Nrg^{*} b$-open and closed function.

A.Maheswari
Department of Mathematics, NGM college, Pollachi, Coimbatore, Tamilnadu, India.
E-mail: vmahes999@gmail.com

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 Article Type : Research Article Title : Equitable and Non-Equitable Zagreb Indices of Graphs Country : India Authors : Akram Alqesmah, Anwar Alwardi and R. Rangarajan

Abstract: The Zagreb indices have been introduced more than forty four years ago by Gutman and Trinajestic as the sum of the squares of the degrees of the vertices, and the sum of the products of the degrees of pairs of adjacent vertices, respectively, \cite{za1}. In this paper, we introduce the first and second equitable and non-equitable Zagreb indices as $M_1^e(G)=\sum_{u\in V(G)}\big[deg_e(u)\big]^2$, $M_2^e(G)=\sum_{uv\in E(G)}deg_e(u)deg_e(v)$, $M_1^{ne}(G)=\sum_{u\in V(G)}\big[deg_{ne}(u)\big]^2$ and $M_2^{ne}(G)=\sum_{uv\in E(G)}deg_{ne}(u)deg_{ne}(v)$, respectively, where $deg_e(u)$ and $deg_{ne}(u)$ denotes the equitable and non-equitable degrees of vertex $u$. Exact values for wheel, firecracker and firefly graph families are obtained, some properties of the equitable and non-equitable Zagreb indices are established.

Keywords: First equitable Zagreb index, Second equitable Zagreb index, First non-equitable Zagreb index, Second non-equitable Zagreb index.

Akram Alqesmah
Department of Studies in Mathematics, University of Mysore, Mysore, India.
E-mail: aalqesmah@gmail.com

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 Article Type : Research Article Title : Edge Domination in Shadow Distance Graphs Country : India Authors : U.Vijayachandra Kumar and R.Murali

Abstract: Let $G$ be a simple connected and undirected graph. The shadow graph of $G$, denoted $D _{2}(G)$ is the graph constructed from $G$ by taking two copies of $G$ namely $G$ itself and $G^ {'}$ and by joining each vertex $u$ in $G$ to the neighbors of the corresponding vertex $u^{'}$ in $G^ {'}$. Let $D$ be the set of all distances between 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 $d(u,v)\in D_s$. In this paper, we define a new graph called the shadow distance graph and determine the edge domination number of the shadow distance graph of the path graph, the cycle graph and the sunlet graph with specified distance sets.

Keywords: Dominating set, Edge domination number, Minimal edge dominating set, Shadow distance graph.

U.Vijayachandra Kumar
School of Physical Science and Computer Applications, REVA University, Bengaluru, India.
E-mail:

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 Article Type : Research Article Title : Generalization of Common Coupled Fixed Point Theorems in Complex Valued b-Metric Spaces Country : India Authors : A.K.Dubey

Abstract: Recently, Azam et al. [1] introduced the complex valued metric space and obtained sufficient conditions for the existence of common fixed points. Rao et al. [20] introduce the notion of complex valued b-metric spaces. In this paper, some common coupled fixed point theorems have been established for a pair of mappings in a complete complex valued b-metric space in view of diverse contractive conditions. Our results extend and improve several fixed point theorems in the literature.

Keywords: Common fixed point, Coupled fixed point, Complete complex valued b-metric space, Complex-valued metric spaces.

A.K.Dubey
Department of Mathematics, Institute of Technology, Bhilai House, Chhattisgarh, India.
E-mail: anilkumardby70@gmail.com

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Abstract: Let $G = (V, E)$ be a simple $(p,q)$ graph and $f:V(G)\rightarrow\{0,1,2,\ldots, p-1\}$ be a bijection. We define $f^{\ast}:E(G)\rightarrow\ \mathbb{N}$ by $f^{\ast}(uv)=(f(u))^{2} + (f(v))^{2} + 2 f(u) \cdot f(v)$, $\forall uv \in E(G)$. If $f^{\ast}$ is injective, then $f$ is called sum perfect square labeling. A graph which admits sum perfect square labeling is called sum perfect square graph. In this paper we prove that several snakes related graphs are sum perfect square.