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On (k) -coloring of generalized Petersen graphs
The chromatic number, ?(G) of a graph G is the minimum number of colors used in a proper coloring of G. In an improper coloring, an edge uv is bad if the colors assigned to the end vertices of the edge is the same. Now, if the available colors are less than that of the chromatic number of graph G, then coloring the graph with the available colors leads to bad edges in G. In this paper, we use the concept of (k)-coloring and determine the number of bad edges in generalized Petersen graph (P(n,t)). The number of bad edges which result from a (k)-coloring of G is denoted by bk(G). 2022 World Scientific Publishing Company. -
On /delta^(k)-colouring of Powers of Paths and Cycles
In animpropervertexcolouringofagraph,adjacentverticesarepermittedto receivesamecolours.Anedgeofanimproperlycolouredgraphissaidtobeabad edge ifitsendverticeshavethesamecolour.Anear-propercolouringofagraphis a colouringwhichminimisesthenumberofbadedgesbyrestrictingthenumberof colour classes that can have adjacency among their own elements. The ?(k)- colouring is anear-propercolouringof G consisting of k givencolours,where1 ? k ? ?(G) ? 1, whichminimisesthenumberofbadedgesbypermittingatmostonecolourclassto have adjacency among the vertices in it. In this paper,we discuss the number of bad edges ofpowers of paths and cycles. Published by Digital Commons@Georgia Southern, 2021. 2021 Georgia Southern University. All rights reserved. -
ON 3-DIMENSIONAL QUASI-PARA-SASAKIAN MANIFOLDS AND RICCI SOLITONS
The purpose of this paper is to study 3-dimensional quasi-para-Sasakian manifolds and Ricci solitons. First, we prove that a 3-dimensional non-paracosymplectic quasi-para-Sasakian manifold is an ?-Einstein manifold if and only if the structure function ? is constant. Further, it is shown that a Ricci soliton on a 3-dimensional quasi-para-Sasakian manifold with ?=constant is expanding. Moreover, we show that if a 3-dimensional quasi-para-Sasakian manifold admits a Ricci soliton, then the flow vector field V is Killing, and the quasi-para-Sasakian structure can be obtained by a homothetic deformation of a para-Sasakian structure. Besides, we study gradient Ricci solitons and prove that if a 3-dimensional non-paracosymplectic quasi-para-Sasakian manifold with ?=constant admits a gradient Ricci soliton, then the manifold is an Einstein one. Also, a suitable example of a 3-dimensional quasi-para-Sasakian manifold is constructed to verify our results. 2022 A. Razmadze Mathematical Institute of Iv. Javakhishvili Tbilisi State University. All rights reserved. -
On an anti-torqued vector field on riemannian manifolds
A torqued vector field ? is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and FischerMarsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative. 2021 by the authors. Licensee MDPI, Basel, Switzerland. -
On an extension of the two-parameter Lindley distribution
AIM: Lindley distribution has been widely studied in statistical literature because it accommodates several interesting properties. In lifetime data analysis contexts, Lindley distribution gives a good description over exponential distribution. It has been used for analysing copious real data sets, specifically in applications of modeling stress-strength reliability. This paper proposes a new generalized two-parameter Lindley distribution and provides a comprehensive description of its statistical properties such as order statistics, limiting distributions of order statistics, Information theory measures, etc. METHODS: We study shapes of the probability density and hazard rate functions, quantiles, moments, moment generating function, order statistic, limiting distributions of order statistics, information theory measures, and autoregressive models are among the key characteristics and properties discussed. The two-parameter Lindley distribution is then subjected to statistical analysis. The paper uses methods of maximum likelihood to estimate the parameters of the proposed distribution. The usefulness of the proposed distribution for modeling data is illustrated using a real data set by comparison with other generalizations of the exponential and Lindley distributions and is depicted graphically. RESULTS/FINDINGS: This paper presents relevant characteristics of the proposed distribution and applications. Based on this study, we found that the proposed model can be used quite effectively to analyzing lifetime data. CONCLUSIONS: In this article, we proffered a new customized Lindley distribution. The proposed distribution enfolds exponential and Lindley distributions as sub-models. Some properties of this distribution such as quantile function, moments, moment generating function, distributions of order statistics, limiting distributions of order statistics, entropy, and autoregressive time series models are studied. This distribution is found to be the most appropriate model to fit the carbon fibers data compared to other models. Consequently, we propose the MOTL distribution for sketching inscrutable lifetime data sets. 2023 DSR Publishers/The University of Jordan. -
On bivariate Teissier model using Copula: dependence properties, and case studies
To precisely represent bivariate continuous variables, this work presents an innovative approach that emphasizes the interdependencies between the variables. The technique is based on the Teissier model and the Farlie-Gumbel-Morgenstern (FGM) copula and seeks to create a complete framework that captures every aspect of associated occurrences. The work addresses data variability by utilizing the oscillatory properties of the FGM copula and the flexibility of the Teissier model. Both theoretical formulation and empirical realization are included in the evolution, which explains the joint cumulative distribution function F(z1,z2), the marginals F(z1) and F(z2), and the probability density function (PDF) f(z1,z2). The novel modeling of bivariate lifetime phenomena that combines the adaptive properties of the Teissier model with the oscillatory characteristics of the FGM copula represents the contribution. The study emphasizes the effectiveness of the strategy in controlling interdependencies while advancing academic knowledge and practical application in bivariate modelling. In parameter estimation, maximum likelihood and Bayesian paradigms are employed through the use of the Markov Chain Monte Carlo (MCMC). Theorized models are examined closely using rigorous model comparison techniques. The relevance of modern model paradigms is demonstrated by empirical findings from the Burr dataset. The Author(s) under exclusive licence to The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2024. -
ON BLOCK-RELATED DERIVED GRAPHS
This paper introduces and analyses the block-degree of a vertex and the cut-degree of a block. The block-degree of a vertex v is the number of blocks containing v. The cut-degree of a block b is the number of cut vertices of G contained in b. The block-degree sequence of cut vertices of the graph and the cut-degree sequence of the graph are defined. A few characterizations of the block-degree and cut-degree sequence of the graph are established. Given a graph, its block graph (B(G)) is a graph where each vertex represents a block, and two vertices are connected if their blocks intersect. The number of cut vertices of B(G) is determined. Further, an investigation is carried out on the traversability of B(G). A block cutpoint graph (BC(G)) of a graph represents a graph where each vertex corresponds to either a block or a cut vertex, and two vertices are connected if one represents a block and the other represents a cut vertex contained within that block. The properties of BC(G) and its iterations are studied. The graph G for which BC(G) is a perfect m-ary tree is characterized. 2024, Canadian University of Dubai. All rights reserved. -
On building up a closer psychic distance as a fundamental ground of relationship between India and Korea: Focusing on Jeonlanam-Do in Korea
The Uppsala model (Johanson and Wiedersheim-Paul, 1975) identifies cultural differences, market attractiveness, and core competence of nations or firms as the key factors affecting international market selection. Among these three factors, psychic distance caused by cultural differences is regarded as the most important factor. However, the psychic distance between India and Korea is not very close. The main objective of this paper is to examine the ways to boost economic relationships between India and Korea by building up a closer relationship of psychic distance. We suggest Jeonlanam-Do including Gwangju Metropolitan City (JDGC) in Korea as a stepping stone to make both countries' psychic distance closer as JD has several common grounds of intangible assets with India which includes its adherence to democracy, human rights and peace; the diverse food culture; and the religious zeal to Buddhism. We propose these common interests as a way to enhance the awareness of national brand 'India' in Korea which will attribute to a strategically developed economic relationship between Korea and India. 2022 Inderscience Enterprises Ltd.. All rights reserved. -
On certain chromatic topological indices of some Mycielski graphs
As a coloring analogue of different Zagreb indices, in the recent literature, the notion of chromatic Zagreb indices has been introduced and studied for some basic graph classes in trees. In this paper, we study the chromatic Zagreb indices and chromatic irregularity indices of some special classes of graphs called Mycielski graphs of paths and cycles. 2020 Yarmouk University. All rights reserved. -
On Certain J-Colouring Parameters of Graphs
In this paper, a new type of colouring called J-colouring is introduced. This colouring concept is motivated by the newly introduced invariant called the rainbow neighbourhood number of a graph. The study ponders on maximal colouring opposed to minimum colouring. An upper bound for a connected graph is presented, and a number of explicit results are presented for cycles, complete graphs, wheel graphs and for a complete l-partite graph. 2019, The National Academy of Sciences, India. -
On certain topological indices of signed graphs
The first Zagreb index of a graph G is the sum of squares of the vertex degrees in a graph and the second Zagreb index of G is the sum of products of degrees of adjacent vertices in G. The imbalance of an edge in G is the numerical difference of degrees of its end vertices and the irregularity of G is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs. 2020 the author(s). -
On equitable chromatic topological indices of some Mycielski graphs
In recent years, the notion of chromatic Zagreb indices has been introduced and studied for certain basic graph classes, as a coloring parallel of different Zagreb indices. A proper coloring C of a graph G, which assigns colors to the vertices of G such that the numbers of vertices in any two color classes differ by at most one, is called an equitable coloring of G. In this paper, we introduce the equitable chromatic Zagreb indices and equitable chromatic irregularity indices of some special classes of graphs called Mycielski graphs of paths and cycles. 2020, SINUS Association. All rights reserved. -
On equitable near proper coloring of graphs
A defective vertex coloring of a graph is a coloring in which some adjacent vertices may have the same color. An edge whose adjacent vertices have the same color is called a bad edge. A defective coloring of a graph G with minimum possible number of bad edges in G is known as a near proper coloring of G. In this paper, we introduce the notion of equitable near proper coloring of graphs and determine the minimum number of bad edges obtained from an equitable near proper coloring of some graph classes. 2024 Azarbaijan Shahid Madani University. -
On equitable near-proper coloring of some derived graph classes
An equitable near-proper coloring of a graph G is a defective coloring in which the number of vertices in any two color classes differ by at most one and the bad edges obtained is minimized by restricting the number of color classes that can have adjacency among their own elements. This paper investigates the equitable near-proper coloring of some derived graph classes like Mycielski graphs, splitting graphs and shadow graphs. Jose S., Naduvath S., 2022. -
On families of graphs which are both adjacency equienergetic and distance equienergetic
Let A(G) and D(G) be the adjacency and distance matrices of a graph G respectively. The adjacency energy or A-energy EA(G) of a graph G is defined as the sum of the absolute values of the eigenvalues of A(G). Analogously, the D-energy ED(G) is defined to be the sum of the absolute values of the eigenvalues of D(G). One of the interesting problems on graph energy is to characterize those graphs which are equienergetic with respect to both the adjacency and distance matrices. A weaker problem is to construct the families of graphs which are equienergetic with respect to both the adjacency and distance matrices. In this paper, we find the explicit relations between A-energy and D-energy of certain families of graphs. As a consequence, we provide an answer to the above open problem (Indulal in https://icgc2020.wordpress.com/invitedlectures, 2020; http://www.facweb.iitkgp.ac.in/rkannan/gma.html, 2020) The Indian National Science Academy 2022. -
On ideal sumset labelled graphs
The sumset of two sets A and B of integers, denoted by A + B, is defined as (formula presented). Let X be a non-empty set of non-negative integers. A sumset labelling of a graph G is an injective function (Formula Presented) such that the induced function (Formula Presented) is defined by (Formula presented). In this paper, we introduce the notion of ideal sumset labelling of graph and discuss the admissibility of this labelling by certain graph classes and discuss some structural characterization of those graphs. 2021 Jincy P. Mathai, Sudev Naduvath, and Satheesh Sreedharan. This is an open access article distributed under the terms of the Creative Commons License, which permits unrestricted use and distribution provided the original author and source are credited. -
On ion transport during the electrochemical reaction on plane and GLAD deposited WO3 thin films
Tungsten oxide thin films were deposited on FTO and Corning glass substrates on Plane and GLAD (75) using DC magnetron sputtering and characterized using SEM, XRD, UVVis spectrophotometer, and Electrochemical analyzer systematically. Further, a comparative analysis was carried out in which it was observed that the result of surface morphology for plane showed the denser and GLAD showed nanopillars deposition. The amorphous nature of the sample was evident from XRD analysis. Optical transmittance was between 87% and 81% for both plane and GLAD. The Electrochemical studies showed the diffusion coefficient of H+ ions are more compared to Li+ ions for both plane and GLAD and Coloration efficiency was calculated at the scan rates of 10, 30, and 50 mV/s at the wavelength of 500 to 600 nm. 2021 -
On J-Colouring of Chithra Graphs
The family of Chithra graphs is a wide ranging family of graphs which includes any graph of size at least one. Chithra graphs serve as a graph theoretical model for genetic engineering techniques or for modelling natural mutation within various biological networks found in living systems. In this paper, we discuss recently introduced J-colouring of the family of Chithra graphs. 2020, The National Academy of Sciences, India. -
On L? (2, 1)-Edge Coloring Number of Regular Grids
In this paper, we study multi-level distance edge labeling for infinite rectangular, hexagonal and triangular grids. We label the edges with non-negative integers. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. We find the edge coloring number of these grids to be 9, 7 and 16, respectively so that we could color the edges of a rectangular, hexagonal and triangular grid with at most 10, 8 and 17 colors, respectively using this coloring technique. Repeating the sequence pattern for different grids, we can color the edges of a grid of larger size. 2019 D. Deepthy et al. -
On l(T, 1)-colouring of certain classes of graphs
For a given set T of non-negative integers including zero and a positive integer k, the L(T, 1)-Colouring of a graph G = (V, E) is a function c: V(G) ? {0, 0, , k} such that |c(u) ? c(v)| ? T if the distance between u and v is 1 and |c(u) ? c(v)| ? 0 whenever u and v are at distance 2. The L(T,1)-span, ?T,1(G) is the smallest positive integer k such that G admits an L(T, 1)-Colouring. In this article we initiate a study of this concept of L(T, 1)-Colouring by determining the value of ?T,1(G) for some classes of graphs and present algorithms to obtain the L(T, 1)-Colouring of paths and stars. 2020 IJSTR.