## Issue 2 − AVolume 4 (2016)

ISSUE 2 − A ISSUE 2 − B ISSUE 2 − C ISSUE 2 − D
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 Article Type : Research Article Title : Vertex Removal and the Edge Domination Number of Graphs Country : India Authors : D.K.Thakkar || B.M.Kakrecha

Abstract: The paper is about the edge domination of the graph. The change in the edge domination number is observed under the operation of vertex removal. Necessary and sufficient condition is proved under which the edge domination number decreases when a vertex is removed from the graph. Calculation of the number of minimum edge dominating sets of the graph is given.

Keywords: Edge domination, edge domination number, edge private neighborhood.

D.K.Thakkar
Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.
E-mail: kakrecha.bhavesh2011@gmail.com

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 Article Type : Research Article Title : One Point Union of $\alpha-$graceful Graphs Country : India Authors : V J Kaneria || Om Teraiya || Parinda Bhatt

Abstract: The notation of $\alpha$-labeling of a graph is the natural generalization of graceful labeling and it helps produce new graceful graphs by some graph operations. In this paper, we have proved that one point union of two $\alpha$-graceful graphs is $\alpha$-graceful under some conditions, one point union of an $\alpha$-graceful graph and a graceful graph is graceful. We have also proved that the consecutive one point union of a finite number of $\alpha$-graceful graphs with a graceful graph by merging one vertex under some conditions is also a graceful graph. These results help enlarge the class of graceful graphs.

Keywords: Graceful graph, $\alpha-$graceful graph, one point union of two graphs.

Om Teraiya
Department of Mathematics, Marwadi Engineering College, Rajkot, India.
E-mail: om.teraiya@gmail.com

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 Article Type : Research Article Title : Analysis of Primes in Arithmetical Progressions $7n + k$ up to a Trillion Country : India Authors : Neeraj Anant Pande

Abstract: Each prime number fits in one of six unique forms $7n + k$. Each form $7n + k$, for $1 \leq k \leq 6$ forms an arithmetical progression; each containing, as assured by Dirichlet's Theorem, infinite number of primes. This work analyzes occurrences of primes in all of these arithmetical progressions from different angles, both 10 power blockwise as well as absolutely in ranges going as high as $1,000,000,000,000$.

Keywords: Arithmetical progressions, block-wise distribution, prime, prime density, prime spacing.

Neeraj Anant Pande
Department of Mathematics \& Statistics, Yeshwant Mahavidyalaya (College), Nanded, Maharashtra, India.
E-mail: napande@gmail.com

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 Article Type : Research Article Title : E--Colouring of a Graph Country : India Authors : D. K. Thakkar || V. R. Dave

Abstract: In this paper we introduce a new colouring of graphs called E -- Colouring. This is not a proper colouring in general. We also define E -- Chromatic Number and E - Homomorphism of a graph. We prove several related results. In particular we prove that for a given E -- chromatic colouring there is a vertex in every colour class which has an equitable neighbour in every other colour class.

Keywords: E-Colouring, E-Chromatic Colouring, E-Chromatic Number, E-Homomorphism, Equitable Independent Set, Maximal Equitable Independent Set, Transversal, Colour Class.

D. K. Thakkar
Department of Mathematics, Saurashtra University, Rajkot, India.
E-mail: dkthakkar1@yahoo.co.in

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 Article Type : Research Article Title : Reliabilty Analysis of Multi-State Series System Country : India Authors : M. Reni Sagayaraj || A. Merceline Anita || A. Chandra Babu || N.Thirumoorthi

Abstract: Reliability Analysis considering Multiple Possible States is known as Multi-State Reliability Analysis. Multi-State System Reliability Models allow both the System and its Components to assume more than two levels of performance. Though Multi-State Reliability Models provide more realistic and more precise representation of Engineering System, they are much more complex and present major difficulties in System definition and performance evaluation. This paper presents a new Systematic Approach for the Reliability Analysis of Multi-State Series System which helps to determine the Expected Throughput and Performance-Free Failure Operation (PFFO) of the System. This Approach is applied to a Mathematical Model dealing with a Particular Population, being categorized into four groups based on their Haemoglobin level and the Health Status of the Population under Study is analyzed.

Keywords: Multi-State System, Series System, Descartes Product, Multi-Series System.

N.Thirumoorthi
Department of Mathematics, Sacred Heart College, Vellore, TamilNadu, India.
E-mail: mercelineanita@gmail.com

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 Article Type : Research Article Title : Soft $g^*s$-closed Mappings in Soft Topological Spaces Country : India Authors : M.Suraiya Begum || M.Sheik John

Abstract: Soft set theory was first proposed by Molodtsov \cite{HG} in 1999 as a general mathematical tool for dealing with not clearly defined objects. Modern topology depends strongly on the ideas of soft set theory.This prompted us to define soft $g^*s$-closed and soft $g^*s$-open mappings on soft topological spaces.

Keywords: Soft Topological Spaces, Soft Closed, Soft Generalized Closed, Soft $g^*s$-closed set, Soft $g^*s$-closed mapping.

M.Suraiya Begum
Department of Mathematics, NGM College, Pollachi, Tamilnadu, India.
E-mail: suraiya0291@gmail.com

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 Article Type : Research Article Title : $\mathcal{I}_{m\omega}$-closed sets Country : India Authors : O. Ravi || K. Indirani || A. R. Thilagavathi

Abstract: In this paper we introduce the notion of $\mathcal{I}_{m\omega}$-closed sets. In Sections 3 and 4, we obtain some basic properties and characterizations of $\mathcal{I}_{m\omega}$-closed sets. In the last section, we define several new subsets in ideal topological spaces which lie between $\star$-closed sets and $\mathcal{I}_\omega$-closed sets.

Keywords: $\mathcal{I}_\omega$-closed set, $\mathcal{I}_{m\omega}$-closed set, m-$\omega$-closed set, m-$\omega$-open set.

O. Ravi
P. M. Thevar College, Usilampatti, Madurai District, Tamil Nadu, India.
E-mail: siingam@yahoo.com

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 Article Type : Research Article Title : The Minimum Paired Dominating Energy of a Graph Country : India Authors : Puttaswamy || S. R. Nayaka

Abstract: Let $G$ be any graph. A dominating set $D$ in $G$ is called a paired dominating set if $\langle D \rangle$ contains a perfect matching. The minimum cardinality of a paired-dominating set is called the paired-domination number, denoted by $\gamma_{pd} (G)$. In this paper, we study the minimum paired dominating energy $E_{pd}(G)$, exact value of $E_{pd}(G)$ is calculated for some standard graphs. Further, the bounds are established for $E_{pd}(G)$.

Keywords: Minimum Paired Dominating set, Minimum Paired Dominating matrix, Minimum Paired Dominating Eigenvalues.

S. R. Nayaka
Department of Mathematics, PES College of Engineering, Mandya, India.
E-mail: nayaka.abhi11@gmail.com

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 Article Type : Research Article Title : Monopoly Free and Monopoly Cover in Graphs Country : India Authors : Ahmed M. Naji and N. D. Soner

Abstract: In a graph $G=(V,E)$, a set $M \subseteq V(G)$ is said to be a monopoly set of $G$ if every vertex $v\in V - M$ has at least $\frac{d(v)}{2}$ neighbors in $M$. The monopoly size $mo(G)$ is the minimum cardinality of a monopoly set among all monopoly sets in $G$. In this paper, we introduce two type sets associated with a monopoly set, monopoly free and monopoly cover sets. A set $F\subseteq V$ is called a monopoly free of $G$ if $F$ does not contain any monopoly set as a subset. The monopoly free set is maximal if for every $v\notin F$ there exists $A\subseteq F$ such that $A \cup \{v\}$ is a monopoly. The cardinality of a largest maximal monopoly free set, denoted $mof(G)$, and is called a monopoly free size of $G$. A set $C \subseteq V$ is called a monopoly cover of $G$ if $C$ contains, at least, one vertex from each monopoly set in $G$. The cardinality of a minimum monopoly cover set in $G$, denoted by $moc(G)$, and is called a monopoly cover size of $G$. We investigate the relationship between these three parameters. Exact values of $mof(G)$ and $moc(G)$ for some standard graphs are found. Bounds for $mof(G)$ and $moc(G)$ are obtained. It is shown that $mof(G) + moc(G) =n$ for every graph $G$ of order $n$. Moreover, it is shown that for every integer $k\geq 3$ there exist a graph $G$ of order $n\geq6$ such that $mof(G)=moc(G)= mo(G)=k$. finally, we characterize all graphs $G$ with $mof(G)= 0, 1$ and $n-2$.

Keywords: Monopoly set, Monopoly free set, Monopoly cover set, Monopoly free size, Monopoly cover size.

Ahmed M. Naji
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru, India.
E- mail: ama.mohsen78@gmail.com

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 Article Type : Research Article Title : Fuzzyfication of Semigroups Country : India Authors : D.D.Padma Priya || G.Shobhalatha || R.Bhuvana Vijaya

Abstract: The motivation mainly comes from the Fuzzification of sets that are of importance and interest in Semigroups, Gamma-Semigroups etc. In this paper we characterize the properties related to Fuzzy sub semigroups and Fuzzy ideals using identities on Semigroups.

Keywords: Fuzzy sub semigroups, Fuzzy (left, right) ideals, Regular Semigroups, Fuzzy bi ideal.

Department of Mathematics, New Horizon College of Engineering, Bangalore, Karnataka, India.

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 Article Type : Research Article Title : Some result on balanced Cordial graphs Country : India Authors : V. J. Kaneria || Jaydev R. Teraiya || Kalpesh M. Patadiya

Abstract: In this paper we have prove that $P(t \cdot C_{4n})$, $C(t \cdot C_{4n})$ and $C_{4n}^ {\ast}$ are balanced cordial graphs under certain condition. A balanced cordial graph is a cordial graph and it satisfies $\vert v_{f}(0) - v_{f}(1) \vert = 0 = \vert e_{f}(0)- e_{f}(1) \vert$ condition.

Keywords: Balanced cordial labeling, cycle of graph, star of graph, path of graph.

V. J. Kaneria
Department of Mathematics, Saurashtra University, Rajkot, India.
E-mail: kaneria_vinodray_j@yahoo.co.in

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 Article Type : Research Article Title : Free Actions of Semiderivations on Semiprime Semirings Country : India Authors : R. Murugesan || K. KanakSindhu || P.Namasivayam

Abstract: Motivated by the study of Laradji and Taheem in $[1]$ in this paper we authors introduce the notion of dependent elements of semiderivations on semiprime semirings. Also in this paper we authors study and investigate dependent elements of semiderivations and proved that semiderivations and related mappings on semiprime semirings are free actions.

Keywords: Semiring, Prime semiring, Semiprime semiring, derivation, semiderivation, dependent element, free action.

R. Murugesan
Thiruvalluvar College, Papanasam, Tamilnadu, India.
E-mail: rmurugesa2020@yahoo.com

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 Article Type : Research Article Title : Generalization of semicompact spaces via bioperations Country : India Authors : R. Nirmala || N. Rajesh

Abstract: In this paper, we introduce $\gamma\vee \gamma^\prime$-semicompact spaces and study some of their properties using the concept of $\gamma\vee \gamma^\prime$-semiconvergence, $\gamma\vee \gamma^\prime$-semifilterbase and $\gamma\vee \gamma^\prime$- semiaccumulation points.

Keywords: $\gamma\vee \gamma^\prime$-semiopen set, $\gamma\vee \gamma^\prime$-semicompact space.

N. Rajesh
Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur, Tamilnadu, India.
E-mail: nrajesh\_topology@yahoo.co.in

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 Article Type : Research Article Title : Indefinitely oscillating functions-Part II Country : Morocco Authors : T. El Bouayachi || N. Yebari

Abstract: This article details a class of indefinitely oscillating functions in $H^{s}$\left[ 0, \infty \right) $. It is a class of functions of the Sobolev space$H^{s} $\left[ 0,\infty \right)$ which have for all $m$ integer one primitive of the order $m$ in the same space.

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T. El Bouayachi
Universit\'{e} Abdelmalek Essa\^{a}di, FST Tanger, D\'{e}partement de Math\'{e}matiques, B.P. 416, Tanger, Morocco.
E-mail:

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 Article Type : Research Article Title : Difference Cordial Labeling of 2-tuple Graphs of Some Graphs Country : India Authors : P L Vihol || P H Shah

Abstract: In this paper, we introduce new notion called 2-tuple graph of an undirected simple graph and we investigate the difference cordiality of 2-tuple graphs of Crown $Cr_n$, Wheel $W_n$, Ladder graph ${P_n} \times {P_2}$, Triangular snake $T_{n}$, Quadrilateral snake $Q_{n}$,Middle graph of path $P_{n}$ and cycle $C_n$ etc.

Keywords: Triangular snake, Quadrilateral snake, Crown, Ladder, Wheel, Path, Cycle.

P L Vihol
Government Engineering College, Rajkot, Gujarat, India.
E-mail: viholprakash@yahoo.com

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 Article Type : Research Article Title : A Study on Skin Cancer Using Fractals Country : India Authors : Dr.V.Shanthoshini Deviha

Abstract: This paper depicts the study of variations in the skin lesion images. The skin cancer images have been analysed for benign, atypical and malignant melanoma types using fractals.

Keywords: Fractals, Skin cancer, Box-Counting dimension and Sausage Dimension.

Dr.V.Shanthoshini Deviha
Department of Mathematics, Trichy Engineeering College, Trichy, Tamilnadu, India.
E-mail: shanthosvijay@gmail.com

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 Article Type : Research Article Title : A Time Truncated Group Sampling Plan For Compound Rayleigh Distribution Country : India Authors : A.R. Sudamani Ramaswamy || N. Sowmya

Abstract: In this paper, group sampling plan is proposed assuming that the life time of the test units follows Compound Rayleigh distribution and the life test is terminated at a prefixed time. The minimum sample size required for ensuring the specified mean life at specified consumer's risk have been determined. The Operating characteristics values for various quality levels are obtained and the results are discussed with the help of tables and examples. The minimum mean ratios are also obtained for a specified level of producer's risk.

Keywords: Acceptance sampling, Compound Rayleigh distribution, Producer's risk, Consumer's risk, Operating Characteristic curve, Truncated life test, Group Sampling Plan.

N. Sowmya
Research Scholar, Department of Mathematics, Avinashilingam Institute For Home Science and Higher Education for Women, Coimbatore, TamilNadu, India.
E-mail: sowmyanagarajan92@gmail.com

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 Article Type : Research Article Title : Heat transfer analysis of MHD stagnation point flow over a stretching sheet with Convective Boundary Conditions Country : India Authors : R. N. Jat || Navin Kumar

Abstract: The problem of a steady two dimensional MHD stagnation point flow and heat transfer analysis over a stretching sheet with CBC and NH is considered. The governing partial differential equation are transformed in to ordinary differential equation using a similarity transformation. Numerical solution are obtained for the skin friction coefficient, the surface temperature, nusselt number as well as the velocity temperature and velocity profiles. The features of the flow and heat transfer characteristics for various values of the Prandtl number, stretching parameter, magnetic parameter and conjugate parameter are analyzed and discussed.

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Navin Kumar
Department of Mathematics, University of Rajasthan, Jaipur, India.

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 Article Type : Research Article Title : Sub-Trident Rankingusing Fuzzy Nmbers Country : India Authors : A.Praveen Prakash || M.GeethaLakshmi

Abstract: This paper deals with the new ranking technique by using Sub-Trident Form along with the help of fuzzy numbers. Here the Sub-Trident Form is obtained from Fuzzy Sub-Triangular Form using fuzzy numbers. The average Sub-Trident Form for each possible path is determined and the minimum of it gives the ranking for the shortest path.

Keywords: Sub-Trident, Ranking, Average, Fuzzy Number, Pascal's Triangle.

M.GeethaLakshmi
Department of Mathematics, KCG College of Technology, Karapakkam, Chennai, Taminadu, India.
E-mail: geetharamon@gmail.com

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