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Issue 3
Volume 5 (2017)

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Authors |
: | S.Saranya, N.Seenivasagan and O.Ravi |

Title |
: | T$_{(1,2)^\star \text{-} \psi}$-spaces |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 1−7 |

**Abstract: ** In this paper, we introduce the notions called T$_{(1,2)^\star \text{-} \psi}$-spaces, gT$_{(1,2)^\star \text{-} \psi}$-spaces and $\alpha$T$_{(1,2)^\star \text{-} \psi}$-spaces and obtain their properties and characterizations.

**Keywords: ** T$_{(1,2)^\star \text{-} \psi}$-space, gT$_{(1,2)^\star \text{-} \psi}$-space, $\alpha$T$_{(1,2)^\star \text{-} \psi}$-space.

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Authors |
: | P.V.Gadagkar |

Title |
: | Initial Value Linear Evolutions Equations Involving Elliptic Operator on the Plane |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 9−12 |

**Abstract: ** A unique solution of an initial value problem involving an elliptic operator on the plane of linear evolution equations is obtained by constructing its Lax pair and performing spectral analysis.

**Keywords: ** Lax pair, evolution equation.

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Authors |
: | Pravin Garg and Shanu Goyal |

Title |
: | Wiener Index of Total Graph of Some Graphs |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 13−24 |

**Abstract: ** Let $G = (V, E)$ be a graph. The \textit{total graph} $T(G)$ of $G$ is that graph whose vertex set is $V \cup E$, and two vertices are adjacent if and only if they are adjacent or incident in $G$. For a graph $G=(V, E)$, the graph $G.S_m$ is obtained by identifying each vertex of $G$ by a root vertex of $S_m$ and the graph $S_m.G$ is obtained by identifying each vertex of $S_m$ except root vertex by any vertex of $G$, where $S_m$ is a star graph with $m$ vertices. In this paper, we consider $G$ as the cycle graph $C_n$ with $n$ vertices and investigate the Wiener index of the total graphs of $C_n.S_m$ and $S_n.C_m$.

**Keywords: ** Topological index, Wiener index, total graph.

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Authors |
: | R.Gopi and C.Suba |

Title |
: | Super Lehmer-3 Mean Labeling of Tree Related Graphs |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 25−27 |

**Abstract: ** Let $f:V(G)\to \left\{1,2,\dots,p+q\right\}$ be an injective function .The induced edge labeling $f^{*} (e=uv)$ is defined by $f^{*} (e)=\left\lceil \frac{f(u)^{3} +f(v)^{3} }{f(u)^{2} +f(v)^{2} } \right\rceil$ (or) $\left\lfloor \frac{f(u)^{3} +f(v)^{3} }{f(u)^{2} +f(v)^{2} } \right\rfloor $, then \textit{f} is called Super Lehmer-3 mean labeling, if $\left\{f(V(G))\right\}U\left\{f(e)/e\in E(G)\right\}=\left\{1,2,3,\dots,p+q\right\}$. A graph which admits Super Lehmer-3 Mean labeling is called Super Lehmer-3 Mean graph. In this paper we prove that $P_{m} \Theta K_{1,n} $, $(P_{m} ,S_{n} )$.

**Keywords: ** Super Lehmer-3 mean graph, $P_{m} \Theta K_{1,n} $, $(P_{m},S_{n} )$.

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Authors |
: | Suhas Patil and Uttam Dolhare |

Title |
: | Unique Fixed Point Theorems for Non-Self Contraction Mapping in Banach Space |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 29−32 |

**Abstract: ** In this paper we prove a unique fixed point theorem for weakly inward condition for non-self contraction mapping of nonempty convex subset of Banach Space.

**Keywords: ** Non-self Mapping, Inwardness Condition, Contraction Mapping, Banach Space.

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Authors |
: | K.Monika and K.Murugan |

Title |
: | Further Odd-Even Sum Labeling Graphs |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 33−37 |

**Abstract: ** A $(p,q)$ graph $G=(V,E)$ is said to be an odd-even sum graph if there exists an injective function $f:V(G)\rightarrow\lbrace\pm 1,\pm 3 \pm 5, ...,\pm (2p-1)\rbrace$ such that the induced mapping $f^{*}:E(G)\rightarrow\lbrace 2,4,6, ...,2q\rbrace$ defined by $f^{*}(uv)=f(u)+f(v)~\forall~uv\in E(G)$ is bijective. The function $f$ is called an odd-even sum labeling of $G$.
In this paper, odd-even sum labeling of subdivision of star $K_{1,n}(n\geq 1)$,Subdivision of bistar $B_{m,n}$,and H-graphs are studied.

**Keywords: ** Odd-even sum graph, odd-even sum labeling.

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Authors |
: | Dhanya Ramachandran and V.Madhukar Mallayya |

Title |
: | Minimizing Property of Two Variable Cubic Spline Interpolation |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 39−42 |

**Abstract: ** In this paper the minimizing property of two variable natural cubic spline interpolation is derived.

**Keywords: ** Cubic spline, Natural Cubic spline, Two variable cubic spline, Minimizing property.

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Authors |
: | Bhagya Jyoti Nath |

Title |
: | A Study on Equivalent Metrics and Equivalent Norms |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 43−46 |

**Abstract: ** In this article we study equivalent metrics and then equivalent norms. Here we study the different conditions for equivalent metrics and equivalent norm and try to compare them. We try to check whether there is any similarity between the conditions for equivalent metrics and equivalent norms and we have come with some conclusion at last.

**Keywords: ** Equivalent metric, Equivalent norm, Strong and weak norm.

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Authors |
: | R.Femina and N.Rajesh |

Title |
: | Properties of $(i, j)$-$\beta$-compact Spaces |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 47−52 |

**Abstract: ** A kind of new $(i, j)$-$\beta$-compactness axiom is introduced in $L$-bitopological spaces, where $L$ is a fuzzy lattice. And its topological properties are systematically studied.

**Keywords: ** $L$-bitopology; $(i, j)$-$\beta$-open set, $(i, j)$-$\beta$-compact set.

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Authors |
: | V.J.Kaneria and H.P.Chudasama |

Title |
: | Graceful Labeling in the Context of Duplication of Some Graph Elements in $\bf K_{m,n}$ |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 53−55 |

**Abstract: ** In this paper, we obtained graceful labeling or $ \alpha $-labeling for some graphs obtained by duplication of some graph elements in the complete bipartite graph $K_{m,n}$.

**Keywords: ** Duplication of a vertex by an edge, duplication of an edge by a vertex, graceful labeling, $ \alpha $-labeling.

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Authors |
: | Bhagya Jyoti Nath |

Title |
: | Applications of Graph Theory in Different Branches of Science |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 57−60 |

**Abstract: ** Graph theory is a branch of mathematics which has wide application in other area of mathematics as well as in other branches of science. It has also significant role in our everyday life. In this article we have discussed the application of graph theory in our daily life and different branches of science such as computer science, operation research, chemistry, physics, engineering etc.

**Keywords: ** Graph, Path, Eulerian Graph, Network.

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Authors |
: | Madhu Jain and Deepa Chauhan |

Title |
: | State Dependent M/$H_{k}$/1/N Queueing System With Service Interruption |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 61−67 |

**Abstract: ** The objective of this paper is to analyze M/$H_{k}$/1 queueing system with setup and unreliable server for both finite and infinite capacity under N-policy. The customers arrive to the system in accordance with Poisson process. The service times follow the k-type hyper exponential distribution. The breakdown and repair times are assumed to follow a negative exponential distribution. The arriving customers may balk, depending upon the number of customers present in the system. The generating function method is used to derive queue size distribution of the number of the customers present in the system. The cost function has been derived in term of cost elements related to different situations in order to determine the optimal operating policy. To explore the effects of various parameters on cost and other indices, the sensitivity analysis is carried out by taking numerical illustrations.

**Keywords: ** N-policy, Unreliable server, M/$H_{k}$/1 queue, Balk, Set up, Loss and delay, Queue size, Generating function.

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Authors |
: | Nabajyoti Bhattacharjee |

Title |
: | Bio-Mathematical Model on Competition between Two Species |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 69−83 |

**Abstract: ** In this paper we shall formulate a mathematical model to study the competition between two species in a given region and at a given interval of time. We shall discuss the nature of competition on the basis of their ability to consume resources in a given period of time that directly depends on population size of a species. Further we shall study and investigate the nature of competition between two species in a long run \textit{i.e.}, when time tends to infinity and to do that we have introduced competition parameter to predict competition and coexistence between both the species when time tends to infinity together with new terminologies like asymptotically superior and asymptotically inferior to define the nature of competition at infinity.

**Keywords: ** Mathematical model, Competition, Superiority Factor, Competition parameter, Competitively superior, Competitively inferior, Competitively analogous, Asymptotically superior, Asymptotically inferior.

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Authors |
: | A.Manshath, V.Balaji and P.Sekar |

Title |
: | On Relaxed Skolem Mean Labling for Five Star |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 85−90 |

**Abstract: ** In this paper, we prove $\ell \le m < n$, the five star ${ K}_{ 1,\ell } \cup { K} _{1,\ell } \cup { K} _{1,\ell } \cup { K} _{1,{ m}} \cup { K} _{1,{ n}}$ is a Relaxed skolem mean graph if $\left|m - n\right| \le 3\ell + 6$ for $\ell= 2, 4, 3, {\dots}$; $m = 2, 3, 4{\dots}$; and $3\ell + m \le n \le 3\ell + m +6$.

**Keywords: ** Graphs, Relaxed Skolem mean graph, star graphs.

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Authors |
: | A.K.Dutta, D.K.Basnet, K.D.Choudhury and S.R.Barbhuiya |

Title |
: | Generalized Doubt Fuzzy Structure of $BG$-algebra |

Volume, Issue, Year |
: | 5(3−A)(2017) |

Pages |
: | 91−104 |

**Abstract: ** In this paper, we introduced the concept of generalized $(\in,\in\vee q_k)$-doubt fuzzy subalgebra and generalized $(\in,\in\vee q_k)$-doubt fuzzy ideal in BG-algebra by using the combined notion of not quasi coincidence $(\overline q) $ of a fuzzy point to a fuzzy set and the notion doubt fuzzy ideals in $BCK/BCI$-algebras. Some characterizations of these generalized doubt fuzzy subalgebra and generalized doubt fuzzy ideal in BG-algebra are derived. We investigated characterizations of $(\in,\in\vee q_k)$-doubt fuzzy subalgebra and $(\in,\in\vee q_k)$-doubt fuzzy ideals by using level sets and $(\in\vee q_k)$-level sets.

**Keywords: ** $BG$-algebra, Fuzzy ideal, Doubt fuzzy ideal, $(\in,\in\vee q_k)$-doubt fuzzy subalgebra,$(\in,\in\vee q_k)$-doubt fuzzy ideal.