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ON NEAR Fk-PERFECT AND DEFICIENT Fk-PERFECT NUMBERS
For a positive integer n, the arithmetic function ?2(n) denotes the sum of squares of all the divisors of n. A positive integer n is called an F-perfect number if ?2(n) ? n2 = 3n. A positive integer n is termed a near F-perfect number if ?2(n) ? n2 ? d2 = 3n, where d is a proper divisor of n. Similarly, n is considered a deficient F-perfect number if ?2(n) ? n2 + d2 = 3n, where d is a proper divisor of n. In this paper, we discuss several characterizations of these numbers, establish their relations with other significant numbers, and generalize the near-perfect and deficient-perfect numbers. 2025, Colgate University. All rights reserved. -
On near-perfect numbers with five prime factors
Let n be a positive integer and ?(n) the sum of all the positive divisors of n. We call n a near-perfect number with redundant divisor d if ?(n) = 2n + d. Let n be an odd near-perfect number of the form n = pa11 ? pa22 ? pa33 ? pa44 ? pa55 where pis are odd primes and ais (1 ? i ? 5) are positive integers. In this article, we prove that 3 | n and one of 5, 7, 11 | n. We also show that there exists no odd near-perfect number when n = 3a1 ? 7a2 ? pa33 ? pa44 ? pa55 with p3 ? {17, 19} and when n = 3a1 ? 11a2 ? pa33 ? pa44 ? pa55 Mathematical and Computational Sciences - Proceedings of the ICRTMPCS International Conference 2023.All rights reserved. -
On path-induced signed graphs
The path decomposition of a graph G is the process of decomposing it into edge-disjoint paths. An induced signed graph is a signed graph formed from an ordinary unsigned graph by assigning signs to its edges according to some protocol. In this paper, we introduce the notion of a path-induced signed graph as an induced signed graph whose edges receive a sign according to whether its end vertices are the end vertices of a path in a path decomposition of G. We also discuss some characteristics of this type of signed graph. The Author(s), under exclusive license to Sapientia Hungarian University of Transylvania 2026. -
On Proper Diameter of Certain Classes of Graphs
An edge coloring of a graph is said to be proper edge coloring if no two adjacent edges receive the same color. A graph G is said to be properly connected if there exists a properly edge colored path between every pair of vertices. For a properly connected graph G with a k-edge coloring c, the proper diameter of a graph, pdiamk (G) is the maximum proper distance between any distinct pair of vertices in G. We investigate the proper diameter of various classes of graphs that are 2-colored and provide bounds on the values of pdiam2(G) for these graphs. Palestine Polytechnic University-PPU 2025. -
ON SECOND HYPER-ZAGREB INDEX OF CORONA PRODUCTS RELATED TO R-GRAPHS
The cognitive and evidential features of the graph discipline are significantly influenced by the implementation of graph operations. Molecular descriptor acts as a fundamental network invariant relevant to a particular molecular structure in the framework of chemical graph theory. The semi-total point graph features the edges of subdivision graph as well as the edges of the original graph. In this paper, we explore combinatorial inequalities associated with the edges, vertices and its corresponding neighborhood notions along with the inclusion of other molecular descriptors in the computations for the determination of exact expressions of second hyper-Zagreb index for certain corona products involving the semi-total point graph. 2023 Academic Publications -
On some classes of equitable irregular graphs
Graph labeling techniques are used by data scientists to represent data points and their relationships with each other. The segregation/sorting of similar datasets/points are easily done using labeling of vertices or edges in a graph. An equitable irregular edge labeling is a function $$f: E(G) \rightarrow N$$ (not necessarily be injective) such that the vertex sums of any two adjacent vertices of $$G$$ differ by at most one, where vertex sum of a vertex is the sum of the labels under $$f$$ of the edges incident with that vertex. A graph admitting an equitable irregular edge labeling is called an equitable irregular graph (EIG). In this paper, more classes of equitable irregular graphs are presented. We further generalize the concept of equitable irregular edge labeling to $$k$$-equitable irregular edge labeling by demanding the difference of the vertex sum of adjacent vertices to be $$k \ge 1$$. The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2021. -
On Some Graphs Whose Domination Number Is thePerfect Italian Domination Number
Perfect Italian Domination (PID) is a vertex labelling of a graph G by numbers from the set such that a vertex in G labelled 0 has a neighbourhood where the summation of the labels of the vertices in it is precisely 2. The summation of labels on the vertices of the graph which satisfy the PID labelling is known as its PID number, and is the minimum possible PID number of a graph G. We find some characterization of graphs for which . We also find a lower bound for |V(G)|, which satisfies the same. Further, we discuss the graphs that satisfies or . A realisation problem is used to prove that PID cannot be bounded by a scalar multiple of the Domination number. The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024. -
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