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On some properties of partial dominating sets
A subset of the vertex set of a graph is a dominating set of the graph if that subset and all the adjacent vertices of that subset form the whole of the vertex set. In case, if a subset and all the adjacent vertices of that subset form part of the whole set, say, for 0 < p < 1, ptimes of the whole vertex set, we say it is a partial domination. In this paper, we explore some of the properties of partial dominating sets with respect to particular values of p. 2020 Author(s). -
On Specific Properties Common to a Graph and its Complement
In this dissertation we study some specific properties common to a graph and its complement. Here we compare the independence number of a graph with the chromatic number of its complement and find some relation between these properties for cycle, path, wheel, Berge graph, star and for some other graphs. We also find the relation between the independence number of a graph and its complement with its order. -
On Statistical Tools in Finding Equitable Antimagic Labeling of Complete Graphs
Graph theory is a branch of mathematics that deals with representation of graphs with vertices and edges. Graph labeling is the assignment of integer labels to either vertices or edges. For a given graph G= (V, E), an edge-weighting is a function f:E(G)?{1,2,3,..,|E(G)|}. For a vertex v of G, let Wf(v) denotes the sum of edge-weights appearing on the edges incident at v under the edge-weighting f. A bijective edge-weighting f of G is said to be an equitable antimagic labeling (EAL) of G if |Wf(u) - Wf(v) | ? 1 for any pair of adjacent vertices u and v of G. A graph admitting an EAL is called an equitable antimagic graph (EAG). In this paper, the characterization of complete graphs Kn, for n? 6 is dealt using an algorithmic approach. 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. -
On StressStrength Reliability Estimation for the Generalized Inverted Exponential Distribution Under Unified Hybrid Censoring
Stressstrength reliability (SSR) analysis plays a fundamental role in reliability engineering, particularly when lifetime data are subject to censoring due to cost or time limitations. In this article, we study the estimation of the reliability parameter (Formula presented.) when the strength (Formula presented.) and stress (Formula presented.) follow the two-parameter generalized inverted exponential distribution (GIED) under a unified hybrid censoring (UHC) scheme, which ensures both a prespecified number of failures and a bounded test duration. Classical inference is developed via maximum likelihood estimation using the EM algorithm, and the corresponding asymptotic confidence intervals are obtained. Bayesian estimation is carried out using MCMC methods under a generalized entropy loss function, along with HPD credible intervals. The UMVUE of (Formula presented.) is also derived for comparison. A Monte Carlo simulation study is conducted to evaluate the performance of the proposed estimators under different censoring scenarios. The results indicate that Bayesian methods, particularly under informative priors, often provide improved estimation accuracy in heavily censored cases. Two real data sets are analyzed to demonstrate the practical applicability of the proposedmethodology. 2026 John Wiley & Sons Ltd. -
On the Anti-Adjacency Spectra of Regular Graphs
For a graph G with vertex set V (G) = {v1, , vn}, the anti-adjacency matrix, denoted by A?(G) is a square matrix of order n with rows and columns indexed by V (G), whose (i, j)? entry (i ? j) is 1, if the vertices vi and vj are not adjacent and 0, otherwise. The diagonal entries of A?(G) is 1. The eigenvalues obtained from A?(G) are called the anti-adjacency eigenvalues of the graph G and the corresponding spectra is called the anti-adjacency spectra, denoted by a-spec(G). In this paper, we discuss the anti-adjacency spectra of connected and disconnected regular graphs and their complement graphs. 2025, SINUS Association. All rights reserved. -
On the compositional and thermal stability of sputter deposited Inconel based multilayer solar absorber coating
A multilayer concept has been used to coat Inconel-718 in the presence of Ar+O2 atmosphere. The coating structure of metal oxide/metal/metal oxide was deposited on stainless steel SS304 substrate with the bottom most oxide layer as non-stoichiometric and the top oxide layer as stoichiometric in nature. This led to the solar absorptance of 0.887 and emissivity of 0.19. The absorption of this multilayer stack was increased by depositing an additional layer of SiO2 which improves the absorptance in the range of 0.940-0.951 without affecting the emittance (0.17 - 0.19). Field Emission Scanning Electron Microscopy analysis was carried out for studying the morphological properties of the coating. The reflectance properties of the coating were analyzed using UV-Vis-NIR spectroscopy and the X-ray diffraction analysis was used for the structural properties. The findings of these studies highlight Inconel's suitability as a solar selective absorber coating. Contrary to high temperature stability of bulk Inconel, thermal stability investigations of the coating reveal its stability only up to 300 C in air for long durations. A detailed metallographic analysis conducted on both the bulk and the metal layer, to compare the intermetallic phases present, revealed the absence of the intermetallic phases in the metallic layer. This absence indeed caused the deterioration in thermal stability of the absorber layer. Further, energy dispersive X-ray analysis revealed that, unlike the bulk material, the sputtered Inconel layer exhibited absence of Nb composition. This absence of Nb is significant, as it contributes to the formation of intermetallic phases, thereby, influencing the observed differences in thermal behavior. 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies. -
On the differential transform method of solving boundary eigenvalue problems: An illustration
The differential transform method (DTM) is a simple technique based on the Taylor series. Applying the DTM, a given linear boundary eigenvalue problem (BEVP) involving ordinary differential equations is converted into a recurrence relation or a system of recurrence relations for the Taylor coefficients. This ultimately leads to the solution of the problem in the form of an infinite power series with an appropriate region of convergence. The present paper aims to apply the DTM in solving a BEVP arising from the DarcyBrinkman convection in a rectangular box subject to general boundary conditions whose vertical sidewalls are assumed to be impermeable and adiabatic. The non-dimensional temperature difference between the plates represented by the DarcyRayleigh number, the eigenvalue of the problem, is obtained as a function of the width of the Bard cell ((Formula presented.) : b is the horizontal wave number) and other parameters using the DTM. The work includes investigation on the convergence of the series solution. The solution by the DTM is compared with that obtained by the MATLAB bvp4c routine and excellent agreement is found thereby establishing the accuracy of theDTM. 2020 Wiley-VCH GmbH -
On the discrete weibull marshallolkin family of distributions: Properties, characterizations, and applications
In this article, we introduce a new flexible discrete family of distributions, which accommo-dates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. Besides, we point out a comprehensive record of some of its mathematical properties. Two distinct estimation methods for parameters estimation and two different methods for constructing confidence intervals are explored for the proposed distribution. In addition, three extensive Monte Carlo simulations studies are conducted to assess the advantages between estimation methods. Finally, the utility of the new model is embellished by dint of two real datasets. 2021 by the authors. Licensee MDPI, Basel, Switzerland. -
On the fully compensated ferrimagnetism in Mn2V1-xCoxAl (x=0, 0.25, 0.5, 0.75, 1) Heusler alloys: An ab initio and neutron diffraction study
We present a detailed ab initio investigation on the interesting Heusler alloys Mn2V1-xCoxAl (x=0,0.25,0.5,0.75,1), which exhibit fully compensated ferrimagnetism with high Curie temperature for x = 0.5. Calculations were performed by incorporating various atomic anti-site disorders, and the minimum energy structure causing the fully compensated ferrimagnetic state with high Curie temperature in Mn2V0.5Co0.5Al was identified to be L 21 with Mn-Co disorder. This L 21 phase, along with the ferrimagnetic interaction among the parallelly coupled (Mn(A)-Mn(C)) and (V-Co) pairs, gives rise to the fully compensated ferrimagnetism in the half-metallic Mn2V0.5Co0.5Al Heusler alloy. Increasing Co concentration in Mn?VAl induces a change in structure from the L 2? phase to the X ? phase. The peculiar spin gapless semiconducting behavior of Mn2CoAl was evident from the ab initio results. Ab initio results have explained the previously reported anomalies in the electrical resistivity of Mn2V1-xCoxAl. Neutron diffraction analysis has confirmed, for the first time, that Mn2V0.5Co0.5Al has a disordered L 2? structure, which agrees with the ab initio results. 2025 Elsevier B.V. -
ON THE GENERALIZED COMPLEMENT OF SOME GRAPHS
In this paper we study the generalized complement of the graph Gm,n = (V, E) for some values of m, n. We study the generalized complement of Gm,n graphs with respect to the equal degree partition. The 2?complement of Gm,n graphs are also determined for m = 2, n is even or odd. In particular, for some values of m, n ? N, we studied the complement of Gm,n graphs with respect to the equal degree partition and the 2?complement of Gm,n graphs. We determine the partitions Pk, k ? N of the vertex set V such that the generalized complement of Gm,n graph is a path graph and a comb graph. 2021, Asia Pacific Academic. All rights reserved. -
On the Hermite and Mathieu Special Characterizations to the Logarithmic ZakharovKuznetsov Equations
In this paper, we find the new travelling wave solutions for several aspects of logarithmic ZakharovKuznetsov (ZK) equations using an efficient technique called the special function method which is composed of Hermite and Mathieu differential equations being novel and special functions. In order to illustrate the efficiency of the projected scheme, we considered four different examples with different cases, namely, logarithmic ZK (log-ZK) equation, logarithmic modified ZK (log-mZK) equation, and logarithmic ZK modified equal width (log-ZK-mEW) equation and logarithmic ZKBenjaminBonaMahony (log-ZKBBM) equation. The behaviour of the obtained results and corresponding consequences are illustrated and captured. Finally, the obtained results confirm that the considered solution procedure can be widely employed to find the solution and also capture some interesting and stimulating consequences. 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited. -
ON THE INDICES OF CERTAIN GRAPH PRODUCTS
Molecular descriptors are numerical graph invariants that are used to study the chemical structure of molecules. In this paper, we determine the upper bound of the Sombor index based on four operations involving the subdivision graph, semi-total point graph, semi-total line graph, and total graph related to the lexicographic and tensor product. The exact expressions of the first reformulated Zagreb index and the second hyper-Zagreb index of the tensor product are formulated on the basis of the four significant graphs. Further, the descriptors for certain standard graphs are obtained and the graphical comparison for the first reformulated Zagreb index has also been illustrated to understand the result better. 2025 University of Isfahan -
ON THE INDICES OF CERTAIN GRAPH PRODUCTS
Molecular descriptors are numerical graph invariants that are used to study the chemical structure of molecules. In this paper, we determine the upper bound of the Sombor index based on four operations involving the subdivision graph, semi-total point graph, semi-total line graph, and total graph related to the lexicographic and tensor product. The exact expressions of the first reformulated Zagreb index and the second hyper-Zagreb index of the tensor product are formulated on the basis of the four significant graphs. Further, the descriptors for certain standard graphs are obtained and the graphical comparison for the first reformulated Zagreb index has also been illustrated to understand the result better. 2025 University of Isfahan -
On the influencing facets of infant mortality in Karnataka: A study based on birth orders
The infant mortality rate (IMR) is used to assess the overall physical health of any community. Reducing this and spreading awareness among people can improve the well-being of society. In India, IMR is high due to the complex and challenging health policies and increased population, but various socio-economic and demographic factors play a significant role in determining the infant mortality rate. This study majorly focuses on identifying the factors influencing infant mortality, and a model has been proposed to estimate the likelihood of an infant's survival in Karnataka. For the empirical analysis, data has been taken from the National Family Health Survey-4 (2015-16), India. It is found that mothers' education and female literacy are the most significant factors affecting the IMR irrespective of the birth order. It is also found that the various socio-economic and demographic factors do not have a significant influence on the survival status of an infant as the birth order increases. Other factors like preceding birth interval, wealth index, caste, and religion also influence infant mortality. Hence, it is suggested that parents should have access to quality education and health facilities near their place of residence to reduce infant mortality at each order of birth. 2025 Author(s). -
On the Investigation of Environmental Effects of ChatGPT Usage via the Newly Developed Mathematical Model in Caputo Sense
This study explores the interconnection between the variables of ChatGPT usage, energy consumption, water consumption, and carbon dioxide CO2 emissions. A new integer and fractional order model using the Caputo derivative is proposed to comprehend the long-term dependencies of these variables. Boundedness, and global and local stability are examined for the fractional order model. The equilibrium points of these variables are shown to determine the stability of the model. The RungeKutta 7 numerical method is employed for the integer order model, whereas the semi-implicit linear interpolation (L1) method is used for the fractional order model. The parameter sensitivity is conducted on the systems parameters to understand the variables impact by varying the relevant parameters for the system. To increase the efficacy of our analysis, we used machine learning approaches to model and predict the dynamics of CO2 emissions, energy and water consumption, and ChatGPT usage. The Prophet ML model stood out among the other methods because it is adept at identifying long-term growth trends, seasonal changes, and the impact of outside variables in intricate time-series data. It is extremely beneficial for research centered on sustainability, where accurate projections are essential for wellinformed decision-making, because it can produce robust, interpretable forecasts against missing values and outliers. Using the Prophet ML model, our research guarantees precise and expandable predictions and provides valuable information that can direct tactics to balance environmental sustainability and technological progress. 2026 by the authors. -
On the k-Forcing Number of Some DS-Graphs
Amos et al. introduced the notion of k-forcing number as a generalization of Zero forcing number and is denoted by Fk(G) where k> 0 is any positive integer, the k -forcing number of a graph is the minimum cardinality among all k -forcing sets of a graph G. In this paper, many bounds for k -forcing number of degree splitting graph DS(G) for different graph classes are found. We evaluate the value of k -forcing number of degree splitting graph of some of the Cartesian product graph for different values of k. Also we observed that for Tur graph Tn , t, upper and lower bound is given by, Fk(Tn , t) ? Fk(DS(Tn , t) ) ? Fk(Tn , t) + 1. 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. -
On the Laplacian energy of interval valued fuzzy graphs
Interval valued fuzzy Laplacian matrix (IVFLM) associated with an interval valued fuzzy graph (IVFG) is studied in this paper. We define spectrum, energy, Laplacian spectrum and Laplacian energy and obtain some bounds for energy and Laplacian energy. 2020 Author(s). -
On the Mass Accretion Rate and Infrared Excess in Herbig Ae/Be Stars
The present study makes use of the unprecedented capability of the Gaia mission to obtain the stellar parameters such as distance, age, and mass of HAeBe stars. The accuracy of Gaia DR2 astrometry is demonstrated from the comparison of the Gaia DR2 distances of 131 HAeBe stars with the previously estimated values from the literature. This is one of the initial studies to estimate the age and mass of a confirmed sample of HAeBe stars using both the photometry and distance from the Gaia mission. Mass accretion rates are calculated from H? line flux measurements of 106 HAeBe stars. Since we used distances and the stellar masses derived from the Gaia DR2 data in the calculation of the mass accretion rate, our estimates are more accurate than previous studies. The mass accretion rate is found to decay exponentially with age, from which we estimated a disk dissipation timescale of 1.9 0.1 Myr. The mass accretion rate and stellar mass exhibit a power-law relation of the form . From the distinct distribution in the values of the infrared spectral index, n2-4.6, we suggest the possibility of difference in the disk structure between Herbig Be and Herbig Ae stars. 2019. The American Astronomical Society. All rights reserved.. -
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On the Maximization of Some Graph Coloring Problems
A graph coloring problem involves labeling the vertices or edges in a graph with newlinecolors or numbers subject to some constraints. The most frequently known graph newlinecoloring problems are the ones that usually minimize the number of colors used in newlinecoloring the vertices or edges. The chromatic number of a graph G, denoted by and#967;(G), is the least number of colors used in a proper coloring of G. The chromatic sum of a graph G, denoted as P(G), was introduced in [1], which is to and the smallest possible coloring sum in a proper coloring of the graph G using natural numbers. Lately, a few studies have endured in a distinct area of the literature where the number of colors used in a graph coloring problem is maximized under certain conditions. Some of these works have applications in network sciences. newlineThe concerned study focuses on the maximization of three dierent edge coloring newlineconcepts, viz., the vertex induced kand#8722;edge coloring, vertex incident kand#8722;edge coloring, newlineand edge incident 2and#8722;edge coloring of a simple connected graph G, where k and#8805; 2. The newlinenumber of colors assigned to the edges of the graph G has been maximized under certain conditions. The vertex induced kand#8722;edge coloring and the vertex incident newlinekand#8722;edge coloring concepts are the generalized version of the edge coloring approach newlineintroduced and studied in [2]. Furthermore, the concept of the achromatic sum of a graph G has also been introduced here. This concept is to and the greatest possible coloring sum of the graph G in an improper edge coloring using natural numbers. An extensive study newlineon three achromatic sums, namely the vertex induced 2and#8722;edge coloring sum, the vertex incident 2and#8722;edge coloring sum, and the edge incident 2and#8722;edge coloring sum are carried out. A few bounds for these parameters on a simple connected graph G and the exact values for some elementary graph classes have been investigated. A few comparative results between some of these parameters have also been obtained.