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Journal articles on the topic 'Perfect square'

Author: Grafiati

Published: 2 June 2025

Last updated: 22 June 2025

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1

Coumbe, D. N. "MAGIC SQUARES OF PERFECT SQUARES AND PELL NUMBERS." JP Journal of Algebra, Number Theory and Applications 63, no. 6 (2024): 587–614. http://dx.doi.org/10.17654/0972555524032.

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Order three magic squares of distinct squared integers are studied. We show that such a magic square is not possible if the smallest entry is the square of a prime number, or unity. A method for generating all arithmetic progressions of three squared integers whose smallest term is the square of a prime or unity is presented via a set of linear transformation matrices involving the Pell numbers.

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2

Chalcraft, Adam. "Perfect Square Packings." Journal of Combinatorial Theory, Series A 92, no. 2 (2000): 158–72. http://dx.doi.org/10.1006/jcta.2000.3058.

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3

Luca, Florian. "Perfect Cuboids and Perfect Square Triangles." Mathematics Magazine 73, no. 5 (2000): 400. http://dx.doi.org/10.2307/2690822.

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4

Luca, Florian. "Perfect Cuboids and Perfect Square Triangles." Mathematics Magazine 73, no. 5 (2000): 400–401. http://dx.doi.org/10.1080/0025570x.2000.11996886.

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5

Banu, P. Shakila, and R. Akilandeswari. "Square Perfect Fuzzy Matching." International Journal of Fuzzy Mathematical Archive 11, no. 1 (2016): 45–52. http://dx.doi.org/10.22457/ijfma.v11n1a7.

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6

Conforti, Michele, Gérard Cornuéjols, and Kristina Vušković. "Square-free perfect graphs." Journal of Combinatorial Theory, Series B 90, no. 2 (2004): 257–307. http://dx.doi.org/10.1016/j.jctb.2003.08.003.

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7

Propp, James G., and Tiberiu V. Trif. "Deriving a Perfect Square: 10706." American Mathematical Monthly 107, no. 9 (2000): 866. http://dx.doi.org/10.2307/2695753.

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8

McDonnell, E. E. "A perfect square root routine." ACM SIGAPL APL Quote Quad 16, no. 4 (1986): 289–94. http://dx.doi.org/10.1145/22008.22050.

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9

Chan, Tsz Ho. "Factors of a perfect square." Acta Arithmetica 163, no. 2 (2014): 141–43. http://dx.doi.org/10.4064/aa163-2-4.

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10

Barbeau, E. J. "Numbers Differing from Consecutive Squares by Squares." Canadian Mathematical Bulletin 28, no. 3 (1985): 337–42. http://dx.doi.org/10.4153/cmb-1985-040-9.

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AbstractIt is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.

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11

Sakai, S. "Unified Formula and Symmetry of Perfect Magic Square." African Journal of Mathematics and Statistics Studies 7, no. 3 (2024): 168–78. http://dx.doi.org/10.52589/ajmss-4h3hywlj.

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Magic squares have long been used in divination and art, due to their magic and wonder. Among them, perfect magic squares are considered valuable as magic squares with special properties, and mathematicians have been interested in them and studied them. However, the achievements that are currently known are how to make a certain perfect magic square, and nothing is known about the number of all perfect magic squares, the unified formula, or the structure. This paper focused on symmetry and clarified the unified formula and structure.

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12

Swarthout, Mary. "Problem Solvers: Perfect-Square Geometry Partners." Teaching Children Mathematics 9, no. 5 (2003): 262–63. http://dx.doi.org/10.5951/tcm.9.5.0262.

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The goal of the “Problem Solvers” department is to foster improved communication among teachers by posing one problem each month for K–6 teachers to try with their students. Every teacher can become an author: pose the problem, reflect on your students' work, analyze the classroom dialogue, and submit the resulting insights to this department. Every teacher can help us all better understand children's capabilities and thinking about mathematics with their contributions to the journal. Remember that even student misconceptions are valuable.

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13

K S P., Sowndarya, and Lakshmi Naidu Y. "Perfect domination separation on square chessboard." Malaya Journal of Matematik 8, no. 4 (2020): 1497–501. http://dx.doi.org/10.26637/mjm0804/0027.

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14

Sabani, Ade Mahmud Eka. "Using the Factor Tree Strategy to Overcome Learning Difficulties for Junior High School Students in Helping to Solve Square Roots." Journal of Medives : Journal of Mathematics Education IKIP Veteran Semarang 7, no. 1 (2023): 1. http://dx.doi.org/10.31331/medivesveteran.v7i1.1850.

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The problem in the research is the difficulty of students in solving problems of the square root of integers (perfect squares) and cube roots of integers (cubic numbers) using an algorithm strategy better known as porogapit. This problem was obtained when giving tutoring to several junior high school students in Purworejo. The algorithm or porogapit strategy was taught when I was in elementary school. Still, as the learning model developed, the strategy could have been more efficient and took a long time. Only a few students remembered the strategy. Today's students even use a trial-and-error strategy: try the same multiplication number one by one or raise the numbers sequentially until the number in question. The trial and error strategy solved only the square root of integers (perfect square). In contrast, when there was a problem with the square root of imperfect integers (irrational squares), they still had difficulty solving it. The results showed that it was easier for students to master the factor tree strategy in solving square root problems than the algorithmic and trial-and-error strategies. Furthermore, the factor tree strategy can assist students in solving the square root problem, both the square root of perfect integers and the square root of imperfect numbers (irrational squares). Keywords: Students Learning Difficulties, Square Root Solutions.

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15

Freiberg, Tristan, and Carl Pomerance. "A note on square totients." International Journal of Number Theory 11, no. 08 (2015): 2265–76. http://dx.doi.org/10.1142/s179304211550102x.

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A well-known conjecture asserts that there are infinitely many primes p for which p - 1 is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes p, q ≤ x for which (p - 1)(q - 1) is a perfect square.

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16

Huxley, M. N., and O. Trifonov. "The square-full numbers in an interval." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (1996): 201–8. http://dx.doi.org/10.1017/s0305004100074107.

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A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

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17

Gopalan, M. A., and V. Geetha. "M-Gonal number ±3 = A Perfect square." International Journal of Mathematics Trends and Technology 17, no. 1 (2015): 32–35. http://dx.doi.org/10.14445/22315373/ijmtt-v17p506.

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18

Martin, Jeremy L., Molly Maxwell, Victor Reiner, and Scott O. Wilson. "Pseudodeterminants and perfect square spanning tree counts." Journal of Combinatorics 6, no. 3 (2015): 295–325. http://dx.doi.org/10.4310/joc.2015.v6.n3.a3.

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19

C.Saranya and Achya B. "Special Diophantine Triples InvolvingSquare Pyramidal Numbers." Indian Journal of Advanced Mathematics (IJAM) 1, no. 2 (2021): 27–29. https://doi.org/10.54105/ijam.B1108.101221.

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In this communication, we accomplish special Diophantine triples comprising of square pyramidal numbers such that the product of any two members of the set added by their sum and increased by a polynomial with integer coefficient is a perfect square

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20

Saranya, Dr C., and B. Achya. "Special Dio 3-Tuples Involving Square Pyramidal Numbers." International Journal for Research in Applied Science and Engineering Technology 10, no. 3 (2022): 891–95. http://dx.doi.org/10.22214/ijraset.2022.40770.

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Abstract: In this communication, we accomplish special dio 3-tuples comprising of square pyramidal numbers such that the product of any two members of the set subtracted by their sum and increased by a polynomial with integer coefficients is a perfect square. Keywords: Special dio 3-tuples, Pyramidal number, perfect square, square pyramidal number.

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21

Saradha, N., and T. N. Shorey. "Squares in blocks from an arithmetic progression and Galois group of Laguerre polynomials." International Journal of Number Theory 11, no. 01 (2014): 233–50. http://dx.doi.org/10.1142/s1793042115500141.

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We investigate when a product of t ≥ 2 terms taken from a set of k successive terms in arithmetic progression is a perfect square. Further, we study the Galois group of Laguerre polynomials. For this, we consider products, which are perfect squares and having terms from two blocks of arithmetic progression.

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22

Nascimento, Rildo Alves do. "Specific properties of exact square roots of nine perfect squares formed by three digits." International Journal of Exact Sciences 1, no. 1 (2024): 1–2. http://dx.doi.org/10.22533/at.ed.153112401073.

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23

Dr., R. Kavitha. "GENERATING INFINITELY MANY PERFECT SQUARES AMONG NARAYANA NUMBERS." International Journal of Current Research and Modern Education (IJCRME) 8, no. 1 (2023): 10–13. https://doi.org/10.5281/zenodo.7764462.

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The Narayana numbers [endif]--> form a triangular array of positive integers, introduced in 1915-1916 by the combinatorialist P. A. MacMahon, and rediscovered in 1955 by the statistician T. V. Narayana<em>. </em>Among Narayana numbers, it turns out that <em>N</em> (1728,28) is a perfect square. A natural question that would arise is whether there exist other values of <em>a </em>such that [endif]-->forms a perfect square? In this paper, we discuss ways of determining infinitely many values of <em>a</em>, for given choice of[endif]-->, such that the Narayana numbers [endif]-->forms a perfect square.

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24

Imam, A.T, M. Balarabe, S. Kasim, and C. Eze. "Perfect Product of two Squares in Finite Full Transformation Semigroup." International Journal of Mathematical Sciences and Optimization: Theory and Applications 11, no. 1 (2025): 107–13. https://doi.org/10.5281/zenodo.15176079.

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 In this paper, we investigate the concept of the perfect product of two squares in the context  of finite full transformation semigroups. We provide a comprehensive analysis of the conditions  under which the product of two idempotent elements in a transformation semigroup forms a  perfect product of two squares. Specifically, we examine the relationship between the kernel and image of idempotents, as well as the interplay between the domain and image of these  transformations. The main result establishes that for two idempotent elements α and β in Tn,  if the domain and image of α and β satisfy certain equivalence conditions, then their product is  a perfect product of two squares. We also explore related properties of disjoint cycles and how these contribute to the structural characteristics of the semigroup. Our findings extend the  existing theory of transformation semigroups and offer valuable insights into the decomposition  of semigroup elements into squares, contributing to the broader field of semigroup theory.

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25

Banks, W. D. "Carmichael Numbers with a Square Totient." Canadian Mathematical Bulletin 52, no. 1 (2009): 3–8. http://dx.doi.org/10.4153/cmb-2009-001-7.

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AbstractLet φ denote the Euler function. In this paper, we show that for all large x there are more than x0.33 Carmichael numbers n ⩽ x with the property that φ(n) is a perfect square. We also obtain similar results for higher powers.

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26

Bhakta, Prateek, and Dana Randall. "Sampling weighted perfect matchings on the square-octagon lattice." Theoretical Computer Science 699 (November 2017): 21–32. http://dx.doi.org/10.1016/j.tcs.2017.01.014.

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27

Jakimczuk, Rafael. "Reciprocal of the square root of a perfect power." International Mathematical Forum 11 (2016): 21–26. http://dx.doi.org/10.12988/imf.2016.5867.

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28

Milanič, Martin, and Oliver Schaudt. "Computing square roots of trivially perfect and threshold graphs." Discrete Applied Mathematics 161, no. 10-11 (2013): 1538–45. http://dx.doi.org/10.1016/j.dam.2012.12.027.

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29

Saranya, C., and B. Achya. "Special Diophantine Triples Involving Square Pyramidal Numbers." Indian Journal of Advanced Mathematics 1, no. 2 (2021): 27–29. http://dx.doi.org/10.35940/ijam.b1108.101221.

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In this communication, we accomplish special Diophantine triples comprising of square pyramidal numbers such that the product of any two members of the set added by their sum and increased by a polynomial with integer coefficient is a perfect square.

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30

Dr., R. Kavitha. "INVESTIGATION OF PERFECT SQUARES AMONG NARAYANA NUMBERS." International Journal of Applied and Advanced Scientific Research (IJAASR) 8, no. 1 (2023): 18–21. https://doi.org/10.5281/zenodo.7729138.

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Narayana numbers named after the Indian mathematician T. V. Narayana (1930–1987) plays a vital role in combinatorial problems. In this paper, we discuss about for what choices of [endif]--> and [endif]--> the Narayana numbers [endif]-->forms a perfect square? In view of answering this question, three cases were discussed by considering odd, even and some specific positive integers. Using combinatorial arguments, divisibility properties, binomial coefficients, we have established three theorems in this paper.

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31

Raju Mudunuri, Lakshmi Narasimha. "Maximizing Every Square Foot: AI Creates the Perfect Warehouse Flow." FMDB Transactions on Sustainable Computing Systems 2, no. 2 (2024): 64–73. http://dx.doi.org/10.69888/ftscs.2024.000198.

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In today’s rapidly evolving industrial landscape, optimizing warehouse operations is paramount for maintaining competitive edge and operational efficiency. This research explores how Artificial Intelligence (AI) can revolutionize warehouse flow by enhancing space utilization, streamlining processes, and reducing operational costs. By integrating AI-driven solutions such as machine learning algorithms, real-time data analytics, and automated systems, warehouses can achieve unprecedented levels of efficiency and accuracy. The proposed method employs a mixed-methods approach, including semi-structured interviews and quantitative analysis of operational data from multiple warehouses that have implemented AI technologies over the past five years. The study uses tools such as regression analysis and hypothesis testing to evaluate the impact of AI on space utilization, inventory accuracy, order picking speed, and labor costs. Results demonstrate significant improvements in space utilization, reduction in labor costs, and enhanced overall productivity. The findings highlight the potential of AI to transform warehouse management, offering a blueprint for future implementations. The paper concludes with a discussion of the implications of AI integration, potential limitations, and avenues for future research.

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32

Sonchhatra, S. G., and G. V. Ghodasara. "Sum Perfect Square Graphs in context of some graph operations." International Journal of Mathematics Trends and Technology 46, no. 2 (2017): 62–65. http://dx.doi.org/10.14445/22315373/ijmtt-v46p512.

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33

T., S. L. Radhika . T. Raja Rani. "Shor's Algorithm – ORACLE Design for Perfect Squares." Journal of Innovation Sciences and Sustainable Technologies 4, no. 1 (2024): 67–75. https://doi.org/10.0517/JISST.2024269298.

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Shor’s algorithm is a groundbreaking quantum computation method engineered to uncover the dual prime factors of an integer formed from these primes. Over the years, numerous research teams have embarked on the journey to harness this algorithm’s power. Yet, their efforts have successfully factorized numbers like 15, 21, 35, and others. This limitation arises from the challenges inherent in quantum hardware, which struggles to manage an abundance of qubits efficiently. The numbers 15, 21, and 35 share a common trait: they can be expressed as the product of two distinct prime factors. In contrast, this paper focuses on the design of an oracle to implement Shor’s algorithm for a unique category of integers - perfect squares. To illustrate, we chose the perfect square 49 as our target and shared our observations and findings regarding this specialized application of Shor’s algorithm.

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34

Yonghe, Deng. "Perfectively Deducing Bessel Mean Square Error Formula." Open Civil Engineering Journal 9, no. 1 (2015): 423–25. http://dx.doi.org/10.2174/1874149501509010423.

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In the survey teaching materials of China, deducing Bessel mean square error formula is all based on survey values with same mathematical expectation. These methods aren’t perfect. So, based on survey values without same mathematics expectation to prove Bessel mean square error formula is very necessary. Therefore, considering different mathematical expectation, it is meaningful that this paper has perfectively deduced Bessel mean square error formula.

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35

Choudhry, Ajai. "Sextuples of integers whose sums in pairs are squares." International Journal of Number Theory 11, no. 02 (2015): 543–55. http://dx.doi.org/10.1142/s1793042115500281.

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This paper is concerned with the diophantine problem of finding six integers such that the sum of any two of them is a perfect square. Till now, only one numerical example of such a sextuple has been published. In this paper, we obtain infinitely many examples of sextuples of integers such that the sum of any two of them is a perfect square. These examples include sextuples which have three or four or five distinct integers as well as sextuples in which all the integers are distinct.

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36

Ram Murty, M., and S. Srinivasan. "Some Remarks on Artin's Conjecture." Canadian Mathematical Bulletin 30, no. 1 (1987): 80–85. http://dx.doi.org/10.4153/cmb-1987-012-5.

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AbstractIt is a classical conjecture of E. Artin that any integer a > 1 which is not a perfect square generates the co-prime residue classes (mod ρ) for infinitely many primes ρ. Let E be the set of a > 1, a not a perfect square, for which Artin's conjecture is false. Set E(x) = card(e ∊ E: e ≤ x). We prove that E(x) = 0(log6 x) and that the number of prime numbers in E is at most 6.

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37

Jakimczuk, Rafael. "Mertens's formula, k-full numbers and sum of two squares." Gulf Journal of Mathematics 17, no. 2 (2024): 263–91. http://dx.doi.org/10.56947/gjom.v17i2.2290.

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In this article we prove generalizations of the well-known Mertens's formula. We also define some arithmetic functions related to the exponents of the primes in the prime factorization of the positive integers and study its average. Finally, we study square-free numbers, s-full numbers and perfect powers not exceeding x which can be represented as sums of two squares.

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38

Peng, Junyao. "The linear combination of two triangular numbers is a perfect square." Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 1–12. http://dx.doi.org/10.7546/nntdm.2019.25.3.1-12.

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39

Jiang, Mei, and Yangcheng Li. "The linear combination of two polygonal numbers is a perfect square." Notes on Number Theory and Discrete Mathematics 26, no. 2 (2020): 105–15. http://dx.doi.org/10.7546/nntdm.2020.26.2.105-115.

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40

Sowndarya, K. S. P., and Y. Lakshmi Naidu. "Perfect Domination for Bishops, Kings and Rooks Graphs On Square Chessboard." Annals of Pure and Applied Mathematics 18, no. 1 (2018): 59–64. http://dx.doi.org/10.22457/apam.v18n1a8.

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41

Osifodunrin, Adegoke. "On the Non-Existence of Difference Sets with Perfect Square Order." British Journal of Mathematics & Computer Science 18, no. 2 (2016): 1–17. http://dx.doi.org/10.9734/bjmcs/2016/9173.

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42

Lee, Huang-Ming, and Jong-Ching Wu. "Temperature Controlled Perfect Absorber Based on Metal-Superconductor-Metal Square Array." IEEE Transactions on Magnetics 48, no. 11 (2012): 4243–46. http://dx.doi.org/10.1109/tmag.2012.2196416.

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43

ZLOSHCHASTIEV, KONSTANTIN G. "MASS OF PERFECT FLUID BLACK SHELLS." Modern Physics Letters A 13, no. 18 (1998): 1419–25. http://dx.doi.org/10.1142/s0217732398001492.

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The spherically symmetric singular perfect fluid shells are considered when their radii are equal to the event horizon (the black shells). We study their observable masses, depending at least on the three parameters, viz. the square speed of sound in the shell, instantaneous radial velocity of the shell at a moment when it reaches the horizon, and integration constant related to surface mass density. We discuss the features of black shells depending on the equation of state.

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44

Gopalan, M., S. Vidhyalakshmi, and N. Thiruniraiselvi. "CONSTRUCTION OF IRRATIONAL GAUSSIAN DIOPHANTINE QUADRUPLES." International Journal of Engineering Technologies and Management Research 1, no. 1 (2020): 1–7. http://dx.doi.org/10.29121/ijetmr.v1.i1.2015.20.

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Given any two non-zero distinct irrational Gaussian integers such that their product added with either 1 or 4 is a perfect square, an irrational Gaussian Diophantine quadruple ( , ) a0 a1, a2, a3 such that the product of any two members of the set added with either 1 or 4 is a perfect square by employing the non-zero distinct integer solutions of the system of double Diophantine equations. The repetition of the above process leads to the generation of sequences of irrational Gaussian Diophantine quadruples with the given property.

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45

Kafi Majumdar, Abdullah Al. "On a sum of squares of integers in arithmetic progression." Journal of Bangladesh Academy of Sciences 45, no. 2 (2022): 241–50. http://dx.doi.org/10.3329/jbas.v45i2.57321.

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This paper derives the conditions under which the sum of squares of (2N+1) natural numbers in the arithmetic progression is a perfect square. It is shown that the problem leads to a Diophantine equation, which in turn indicates that there is, in fact, an infinite number of such numbers. Some particular cases are investigated. J. Bangladesh Acad. Sci. 45(2); 241-250: December 2021

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46

Jóźwik, Renata. "The problem of urban redevelopment of post-industrial King’s Cross central area in London." Budownictwo i Architektura 17, no. 1 (2018): 063–69. http://dx.doi.org/10.24358/bud-arch_18_171_08.

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The paper presents the strategy of redevelopment of the post-industrial King’s Cross area in London – one of the biggest European investments in the last years (the surface of approx. 67 acres), which could be considered the perfect case study for similar works.The author described in details the investment process, as well as the principles and effects of functional and spatial changes that have led to creation of the new system of open space, adaptation of selected post-industrial buildings for new functions (also the buildings proclaimed as a monument) and also fostering a new urban dimension to the wastelands. Detailed architectural issues are presented on the example of 3 squares: Granary Square, Pancras Square and King’s Cross Square. Field research was realized from 2011 to 2016.

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47

Li, Bingzhen, Yuhua Chen, Qingqing Wu, et al. "Ultrathin Narrowband and Bidirectional Perfect Metasurface Absorber." Coatings 13, no. 8 (2023): 1340. http://dx.doi.org/10.3390/coatings13081340.

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The conventional design approaches for achieving perfect absorption of electromagnetic (EM) waves using metasurface absorbers (MSAs) are limited to absorbing waves in one direction while reflecting waves in the other. In this study, a novel ultrathin narrowband MSA with bidirectional perfect absorption properties has been proposed, based on a tri-layer metal square-circular-square patch (SCSP) structure. The simulation results demonstrate that the proposed MSA exhibits a remarkable absorbance of 98.1%, which is consistent with the experimental and theoretical calculations. The equivalent constitutive parameters that were retrieved, as well as the simulated surface current and the power loss density distributions, reveal that the perfect absorption of the designed MSA originates from the fundamental dipolar resonance. Furthermore, the proposed MSA demonstrates stable wide-angle absorption properties for both transverse electric (TE) and transverse magnetic (TM) waves under various oblique incidence angles. The absorption characteristics of the MSA can be fine-tuned by adjusting the structural parameters. Additionally, the proposed MSA boasts excellent ultrathin thickness, bidirectional, polarization-insensitive, and wide-angle properties, making it highly suitable for a range of potential applications such as imaging, detection, and sensing.

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48

Saranya, C., and B. Achya. "Special Diophantine Triples InvolvingSquare Pyramidal Numbers." Indian Journal of Advanced Mathematics 1, no. 2 (2021): 27–29. http://dx.doi.org/10.54105/ijam.b1108.101221.

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In this communication, we accomplish special Diophantine triples comprising of square pyramidal numbers such that the product of any two members of the set added by their sum and increased by a polynomial with integer coefficient is a perfect square.

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49

Appleby, D. M., Ingemar Bengtsson, Stephen Brierley, Asa Ericsson, Markus Grassl, and Jan-Ake Larsson. "Systems of Imprimitivity for the Clifford group." Quantum Information and Computation 14, no. 3&4 (2014): 339–60. http://dx.doi.org/10.26421/qic14.3-4-9.

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It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of $k$-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

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50

Gandal, Ganesh, R. Mary Jeya Jothi, and Narayan Phadatare. "On very strongly perfect Cartesian product graphs." AIMS Mathematics 7, no. 2 (2022): 2634–45. http://dx.doi.org/10.3934/math.2022148.

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Abstract:

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>

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