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TitleOn Characterization of Quadratic Exponential Invertible Graphs
Vol.47(6) (2023) page: 733-741
Author(s): S. Ali, M.K. Mahmood and K.P. Shum  

In this paper, we investigate the notion of quadratic exponential invertible graphs whose vertex set is reduced residue system $mod~n$ and there will be an edge between $x$ and $y$ such that $x^{2^{\alpha}}\equiv y^{2^{\alpha}}(mod~n)$ for some positive integer $\alpha$. The proposed graph is completely characterized for each positive integer $n$ and also, we find the class of integers in which quadratic exponential invertible graphs are isomorphic to each other. Moreover, the class of those integers is investigated in which the proposed graph is a complete graph.

Keywords: Invertible elements; Quadratic exponential invertible graphs; Complete graphs.
Size:198KB   download
TitleThe Conjugacy Classes Ranks of the Alternating Simple Group $A_{11}$
Vol.47(6) (2023) page: 743-753
Author(s): A.B.M. Basheer, M.J. Motalane and T.T. Seretlo  

Let $G$ be a finite group and $X$ be a conjugacy class of $G.$ The $rank$ of $X$ in $G$, denoted by $rank(G~{:}~X)$, is defined to be the minimum number of elements of $X$ generating $G.$ We investigate the ranks of the alternating group $A_{11}.$ We use the structure constants method to determine the ranks of all the non-trivial classes of the group $A_{11}.$

Keywords: Conjugacy classes; Rank; Generation; Alternating simple group.
Size:134KB   download
TitleNeighborhood Pseudo Achromatic Number of a Complete p-Partite Graph
Vol.47(6) (2023) page: 755-768
Author(s): R. Divya and M. Jayalakshmi   

A pseudo coloring of the vertices of a graph $G$, is a coloring assigned to the vertices of $G$ such that adjacent vertices can receive the same colors. A pseudo complete coloring of a graph $G$ is a pseudo coloring of the vertices of $G$, such that for any pair of distinct colors, there is at least one edge whose end vertices are colored with this pair of colors. The maximum number of colors used in a pseudo complete coloring of $G$ is called the pseudo achromatic number, denoted by $\psi_{s}(G)$. A neighborhood pseudo chromatic number of a graph $G$, denoted by $\psi_{nhd}(G)$, is the maximum number of colors used in a pseudo coloring of $G$, such that every vertex has at least two vertices in its closed neighborhood receiving the same color. Here, we defined a new coloring parameter called neighborhood pseudo achromatic number of a graph $G$ and obtained neighborhood pseudo achromatic number of a complete $p$-partite graph by analysing various possibilities.

Keywords: Neighborhood pseudo coloring; Achromatic number; Pseudoachromatic number; Neighborhood pseudo achromatic number; Complete $n$-partite graph.
Size:235KB   download
TitleThree Parameters Fuzzy Sets Framework: Applications in Non Associative Algebraic Structures
Vol.47(6) (2023) page: 769-789
Author(s): M. Khan, S. Khan, Zulfiqar, S. Anis, S. Iqbal and S.Z. Song  

In this paper, we introduce the concept of three parameters-fuzzy sets. We define three parameters fuzzy ideals in non associative algebraic structures namely Abel Grassman groupoids (AG-groupoids). We explored several properties of an intra-regular AG-groupoid using three parameters-fuzzy AG-subgroupoids and three parame\-ters-fuzzy right ideals.

Keywords: AG-groupoid; Completely regular; $(\in _{\gamma },\in _{\gamma }\vee q_{\delta }^{\eta })$-fuzzy ideals and $(\in _{\gamma },\in _{\gamma }\vee q_{\delta }^{\eta })$-fuzzy ideals.
Size:188KB   download
TitleExtended Hyperbolic Function and its Properties
Vol.47(6) (2023) page: 791-804
Author(s): M. Menon, E. Mittal and R. Gupta   

Aim of this paper is to introduce extended hyperbolic function by using a modified extension of beta function [12] and to establish new properties like integral representation, Mellin transform and many more. Furthermore, we apply Prabhakar fractional integral operator, Caputo-Fabrizio operator and Atangana-Baleanu operator on it. Other than this, we present a graphical representation of the extended hyperbolic function with different values of $\alpha$ also a graphical comparison between Caputo-Fabrizio operator and Atangana-Baleanu operator of hyperbolic function for different values of $r$.

Keywords: Extended beta function; Extended Hyperbolic function; Prabhakar function; Mittag-Leffler function; Caputo-Fabrizio derivative; Atangana-Baleanu derivative.
Size:238KB   download
TitleReproducing Kernel for Periodic Boundary Conditions
Vol.47(6) (2023) page: 805-818
Author(s): G. Patel and K. Patel  

In this paper, we introduced a reproducing kernel space which is a particular class of Hilbert space. We discuss various properties of the reproducing kernel. In particular, our aim is to construct kernel in reproducing kernel Hilbert space of the specific function space (Sobolev space) with the improved inner product and norm. Also, we derive the reproducing kernel for periodic boundary conditions.

Keywords: Inner product; Hilbert space; Reproducing kernel Hilbert space; Reproducing kernel; Periodic boundary conditions.
Size:146KB   download
TitleInclusion Relations of Various Subclasses of Harmonic Univalent Mappings and k-uniformly Harmonic Starlike Functions
Vol.47(6) (2023) page: 819-834
Author(s): S. Porwal and N. Magesh  

The purpose of the present paper is to obtain inclusion relations between various subclasses of harmonic univalent mappings by applying a convolution operator involving generalized Wright functions. To be more precise, we investigate such connections with Goodman-R{\o}nning-type harmonic univalent functions, $k$-uniformly harmonic convex functions and $k$-uniformly harmonic starlike functions in the open unit disc $\mathbb{U}$. Some of our results generalize and correct the results of Maharana and Sahoo [11].

Keywords: Analytic; Harmonic functions; Harmonic convex functions; Harmonic starlike functions; Wright function.
Size:153KB   download
TitleOn Metric Dimension of Circulant Graph $C_n(1, 2)$ Joining $n$-paths
Vol.47(6) (2023) page: 835-843
Author(s): S.K. Sharma and V.K. Bhat   

Let $H=H(V, E)$ be a graph. A subset of vertices $M$ in $V(H)$ is said to be a resolving set (or metric generator) for $H$ if every $y, z\in V(H)$ with $y\neq z$, there exists a vertex $a\in M$ such that $d(a, y)\neq d(a, z)$. A metric generator containing a minimum number of vertices is called a metric basis for $H$ and the cardinality of this metric basis is the metric dimension of $H$, denoted by $dim(H)$. Let $C^{q}_{n}(1, 2)$ be a graph obtained from the circulant graph $C_{n}(1, 2)$ by joining $n$-paths of length $q$ at each vertex of the graph $C_{n}(1, 2)$. In this work, we show that the metric dimension of the graph $C^{q}_{n}(1, 2)$ is three when $n\equiv0, 2, 3 \ mod(4)$ and four when $n\equiv1\ mod(4)$.

Keywords: Circulant graph; Metric dimension; Resolving set; Pendant vertices; Pendant edges.
Size:162KB   download
TitleGeometry in A-Metric Space
Vol.47(6) (2023) page: 845-853
Author(s): B.V. Subba Rao and P. Yedlapalli  

In this paper, we study the geometry of an A-metric space of Subba Rao [15], by introducing the concepts of metric betweenness and its properties $t_{1}$, $t_{2}$, B-linearity and D-linearity. It is proved that there do not exist equilateral triangles in any A-metric space $\left(A,A,d \right)$, where A is any representable autometrized algebra satisfying $\left(R\right)$ and$\left(S\right)$. It is also proved that any A-metric space is Ptolemaic.

Keywords: Autometrized algebra; A-metric space; Boolean metric space; B-Linearity; Contraction mapping; D-Linearity; G-metric space and lattice.
Size:98KB   download
TitleOn Pseudo-Slant Submanifolds of Conformal Kenmotsu Manifoldsmetric
Vol.47(6) (2023) page: 855-867
Author(s): Venkatesha, N. Srikantha and M.S. Siddesha  

In this paper, we study the pseudo-slant submanifolds of conformal Kenmotsu manifolds. Some results on submanifolds of conformal Kenmotsu manifolds with parallel canonical structures are obtained. Finally we discuss integrability of the distributions embeded in the definition of pseudo-slant submanifolds of conformal Kenmotsu manifolds.

Keywords: Conformal Sasakian manifolds; Integrability; Parallel; Pseudo-slant submanifolds.
Size:117KB   download
TitleA Study of the Green's Relations on n-Potent Semirings
Vol.47(6) (2023) page: 869-879
Author(s): J.L. Wang, Y. Shao and X.L. Xian  

In this paper, we study the semirings which satisfy the identities $x^{n} \approx x$, $(2^{n}-1)x \approx x $, $(x+y)^{n-1}\approx x^{n-1}+y^{n-1}$ and $(xy)^{n-1}\approx x^{n-1}y^{n-1}$. We give the characterizations of the binary relations $\mathop {\cal L}^ \cdot \wedge \mathop {\cal D}^ + $, $\mathop {\cal L}^ \cdot \wedge \mathop {\cal L}^ + $, $\mathop {\cal L}^ \cdot \wedge \mathop {\cal R}^ + $ and $\mathop {\cal L}^ + \wedge \mathop {\cal D}^ \cdot $, and obtain the sufficient and necessary conditions which make these binary relations be congruences. Finally, we show that the classes of semirings which can be determined by the above congruences are indeed varieties of semirings.

Keywords: Green's relations; Congruences; Semirings; Varieties of semirings.
Size:123KB   download
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