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On estimation of extropy for non-negative data with application on uniformity testing
{Poisson weights-based density estimator is used to estimate the extropy function to the non-negative data}. The traditional class of nonparametric extropy estimators, typically constructed using kernel density estimators with symmetric kernels, is not well suited for non-negative data. To address this limitation, we propose two Poisson-weights-based density estimators that are naturally adapted to the non-negative domain. The asymptotic properties of the proposed estimators are rigorously established, providing theoretical support for their use. A comprehensive simulation study demonstrates that both estimators outperform their conventional kernel-based counterparts in terms of bias and mean squared error. Furthermore, we introduce uniformity tests based on extropy and obtain their critical values through simulation. The practical utility of the proposed methods is illustrated through analyses of real data sets. 2025 Informa UK Limited, trading as Taylor & Francis Group. -
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 Improving Quality of Experience of 4G Mobile Networks A Slack Based Approach
This paper analyses Indias four top 4G Mobile network Providers with respect to five key user experience metrics Video, Games, Voice app, Download speed and Upload speed. Results using Data Envelopment Analysis show Airtel and Vodafone-Idea performing with maximum relative efficiency with respect to these metrics, while BSNL and Jio closely follow them. Further analysis using the Slack Based Measure shows where and by how much BSNL and Jio need to improve to perform at par with Airtel and Vodafone-Idea. On certain variables, for instance Voice app, BSNL and Jio perform well, with no need for improvement. On the contrary, for Upload and Download speed experiences, both BSNL and Jio lag. For Video and Games, there is still scope for improvement, although both these players are reasonable in their performance. Thus, this analysis provides an accurate and optimal benchmark for each variable whose user experience has been evaluated. 2021, Springer Nature Switzerland AG. -
On interval valued fuzzy graphs associated with a finite group
We associate a particular type of interval-valued fuzzy graph(IVFG) called interval-valued fuzzy identity graph(IVFIG) with every finite group and study its various properties. We show that IVFIG associated with a finite group is not unique. We also show that every IVFIG associated with a finite group is a strong IVFG. It does not contain any feeble or weak arcs. Further, it is strongly connected. We prove that the IVFIG associated with a finite group in which every element is self inversed is an interval-valued fuzzy tree and the IVFIG of Zn (n is odd) under addition modulo n is the disjoint union of interval-valued fuzzy cycles. 2020 Author(s). -
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 k-Facile Perfect Numbers
For a positive integer n, let ?(n) denote the sum of all positive divisors of n. Then n is said to be a k-facile perfect number if ?(n) = 2n + d1d2 dk, where 1 < d1, d2,, dk < n are distinct divisors of n. This paper characterizes k-facile perfect numbers and establishes their relationships with other special numbers. The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025. -
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(2, 1)-ORDER SUM SIGNED GRAPH OF A FINITE GROUP
In this paper, we have constructed a color-induced signed graph of an algebraic graph, called the L(2, 1)-order sum signed graph of a group. Based on the nature of the group, the L(2, 1)-span of the order sum graph is obtained and the structural aspects of thus obtained L(2, 1)-order sum signed graph such as planarity, chordality, etc. have been investigated. We have also defined an automorphism which turns out to be the only possible automorphism on the graph and have investigated the structural aspects of the graph such as edge transitivity and vertex transitivity. Further, a line-signed graph of L(2, 1)-order sum signed graph, which is a line graph with a signing protocol defined for the edges, has also been introduced. We have also explored the regularity of the line-signed graph. 2025 Sciendo. All rights reserved. -
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. -
On Leech labelings of graphs and some related concepts
Let G=(V,E) be a graph and let f:E?{1,2,3,} be an edge labeling of G. The path weight of a path P in G is the sum of the labels of the edges of P and is denoted by w(P). The path number of G, tp(G) is the total number of paths in a graph G. If the set of all path weights S in G with respect to the labeling f is {1,2,3,,tp(G)}, then f is called a Leech labeling of G. A graph which admits a Leech labeling is called a Leech graph. Leech index is a parameter which evaluates how close a graph is towards being Leech. In this paper, the path number of the wheel graph Wn is obtained. We also determine a bound for the Leech index of Wn and a subclass of unicyclic graphs. A python program that gives all possible Leech labelings of a cycle Cn for n?3, if it exists, is also provided. 2023 Elsevier B.V. -
On m-quasi Einstein almost Kenmotsu manifolds
In this article, we consider m-quasi Einstein structures on two class of almost Kenmotsu manifolds. Firstly, we study a closed m-quasi Einstein metric on a Kenmotsu manifold. Next, we proved that if a Kenmotsu manifold M admits an m-quasi Einstein metric with conformal vector field V, then M is Einstein. Finally, we prove that a non-Kenmotsu almost Kenmotsu (?,?)' -manifold admitting a closed m-quasi Einstein metric is locally isometric to the Riemannian product Hn+1Rn, provided that ?-?(2n+m)/2m = 1. 2021 Universita degli Studi di Parma. All rights reserved. -
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.