EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6095 ISSN 1307-5543 – ejpam.com Published by New York Business Global Interval Valued Spherical Fuzzy Matrix in Decision Making Riyaz Ahmad Padder1,∗, Taghreed Alqurashi2, Yasir Ahmad Rather1, Shilpa Malge3 1 Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Jalandhar, Punjab, India 2 Mathematics Department, Faculty of Science, Al-Baha University, 65779-7738, Albaha City, Kingdom of Saudi Arabia 3 Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University) (SIU), Pune, India Abstract. Recent advancements have demonstrated the potential to augment matrix theory with fuzzy, intuitionistic fuzzy, picture fuzzy, interval-valued picture fuzzy matrix concepts for enhanced decision-making applications. We introduce the interval valued spherical fuzzy matrix, extending the spherical fuzzy matrix, to effectively represent and manipulate uncertain and vague information with enhanced flexibility. This paper establishes definitions and theorems for Interval-Valued spherical fuzzy matrices. We develop methods for computing determinant and adjoint, and develop algorithms using composition functions to determine the greatest and least eigenvalue interval valued spherical fuzzy sets and create a flow chart to depict the procedure. In this paper, a new distance measure has been proposed and is to be proved valid by satisfying all the conditions of the distance metric. In addition, an application of interval-valued spherical fuzzy matrices to deal with decision-making problems is presented. 2020 Mathematics Subject Classifications: 03E72, 15B15, 90B50 Key Words and Phrases: Interval valued spherical fuzzy sets, Interval valued spherical fuzzy matrix, Least eigenvalue, Greatest eigenvalue Abbreviations IVSFS Interval valued spherical fuzzy set IVSFSs Interval valued spherical fuzzy sets IVSFM Interval valued spherical fuzzy matrix IVSFMs Interval valued spherical fuzzy matrices ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6095 Email addresses: riyaz.28709@lpu.co.in (R.A. Padder), talqorashi@bu.edu.sa (T. Alqurashi), yarmth111@gmail.com (Y.A. Rather), shilpam@sitpune.edu.in (S. Malge) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 2 of 24 EIVSFS Eigen interval valued spherical fuzzy set GEIVSFS Greatest Eigen Interval valued spherical fuzzy set LEIVSFS Least Eigen Interval valued spherical fuzzy set IVSFR Interval valued spherical fuzzy relation AO Administrative officer AOs Administrative officers Government Govt. Governments Govts. DOC Degree of closeness DOCs Degree of closenesses 1. Introduction Uncertainty pervades our lives in our daily and scientific pursuits, affecting more or less most applications of interest. The medical, engineering, industrial, economic, and business domains present themselves as natural battlefields for making decisions in the presence of uncertainty when real-world applications are dealt with. In trying to overcome these, the existing methodologies are grossly inadequate. Addressing this imperative, Zadeh [1] introduced a theory called fuzzy set that is a milestone in studying certain varieties of un- certainty with which classical modes of study cannot cope or have utterly failed. Fuzzy set theory and its generalizations have resulted in great mathematical applications to almost all real-life problems involving uncertainty. Many extensions and modifications of fuzzy set theory have been developed to handle various forms of uncertainty. These include vague sets, rough sets, intuitionistic fuzzy sets, soft sets, Pythagorean fuzzy sets, picture fuzzy sets, and other advanced generalizations. Kim and Roush [2] introduced the concept of fuzzy matrices, which play a vital role, in various fields of Science and Engineering, addressing problems involving various types of uncertainties [3]. Hashimoto [4] examined the convergence of power of transitive fuzzy ma- trix. Since then, extensive work has been done on fuzzy matrices. However, fuzzy matrix focus membership value only. Atanassov [5] extended the idea of fuzzy sets by introducing the of concept intuitionistic fuzzy sets, Which add a new dimension called the hesitancy de- gree. While fuzzy sets have membership and non membership degrees, intuitionistic fuzzy sets add hesitancy degree, which quantifies the level of uncertainty or ambiguity associated with an element’s classification. This Additional dimension (hesitancy degree) allows in- tuitionistic fuzzy sets to solve complex problems involving imprecise or vague information. Since their introduction, intuitionistic fuzzy sets have been widely discussed and applied across various domains, including decision-making [6, 7], pattern recognition [8], and med- ical diagnosis [9, 10] where uncertainty plays a important role in analysis. El-Morsy [11] proposed an innovative method utilizing Pythagorean fuzzy numbers to evaluate the risked return rate, portfolio risk, and expected return rate. Atanassov and Gargov [12] expanded the concept of Intuitionistic Fuzzy Sets (IFS) to include interval-valued intuitionistic fuzzy sets, where interval numbers replace exact numbers, offering greater flexibility in defining membership degrees for elements. The versatility and effectiveness of interval-valued intu- R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 3 of 24 itionistic fuzzy sets have been demonstrated across various domains, as evidenced by nu- merous studies [13–16]. Additionally, pal [17] introduced intuitionistic fuzzy matrices, and several properties of these matrices were explored. Investigations performed by Bhowmik and Pal [18] focused on the convergence of max-product powers in intuitionistic fuzzy ma- trices. Similarly, Pradhan and Pal [19] examined the mean powers of convergence within these matrices. Additionally, the study of power convergence and the canonical structure of sss-transitive intuitionistic fuzzy matrices has been conducted [20]. Several researchers [21–27] have contributed to advancing the field of intuitionistic fuzzy matrices, achieving numerous significant results that prove useful for addressing uncertainty in real-life prob- lems. However, despite their wide applicability, intuitionistic fuzzy sets fall short when it comes to handling contradictory scenarios in many practical applications. For instance, consider a voting system where outcomes can be categorized as voting for, abstaining, vot- ing against, or refusing to vote. To address such complexities, Cuong and Kreinovich [28] introduced the concept of picture fuzzy sets. These sets encompass three degrees of mem- bership: membership, neutral membership, and non-membership. This extension allows picture fuzzy sets to generalize intuitionistic fuzzy sets, enabling them to capture more nuanced, conflicting, and ambiguous information. Furthermore, Cuong and Kreinovich defined foundational operational laws for picture fuzzy sets and established several of their properties. Garg [29] introduced the weighted average and geometric aggregation operators for picture fuzzy numbers and applied them to decision-making problems. Mul- tivariable Hermite and Apostol-type Frobenius–Genocchi polynomials enhance spherical fuzzy matrices by refining uncertainty modeling in decision-making processes [30]. The novel bivariate 2D-q Hermite polynomials can be used to define complex membership func- tions in spherical fuzzy matrices, enhancing multidimensional uncertainty representation in decision-making[31]. Ganie [32] proposed a new distance measure for picture fuzzy sets and demonstrated the application of the proposed distance measure in pattern recogni- tion. Dogra and Pal [33] developed the concept of picture fuzzy matrices and explored their properties. Thereafter, many authors [34, 35] extended the study of picture fuzzy matrices. While picture fuzzy sets have a variety of dimensions, they are also bound by the same constraint as intuitionistic fuzzy sets the sum of their three parameters, membership, neutral membership, and non-membership grades must be ≤ 1. To break through this bar- rier, Mahmood et al. [36] extended the picture fuzzy set structure into a T-Spherical Fuzzy Set, which lets its membership parameters range over an enlarged range of values. For example, Kifayat et al.[37] did a geometrical comparison for fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and T-Spherical fuzzy sets. Silamarasan [38] introduced the spherical fuzzy matrix and derived some of its properties with the help of matrix operations. In this paper, we introduce interval valued spherical fuzzy matrix extending spherical fuzzy matrix (SFM), to effectively represent and manipulate uncertain and vague information with enhanced flexibility. We develop algorithms to find the greatest eigen vector and least eigen vector for an interval valued spherical fuzzy matrix using composition and propose a new distance measure, verified its properties. The application based on the decision making problem is presented with the help of an example. R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 4 of 24 The main research gap between SFMs and IVSFMs lies in their ability to model uncer- tainty and flexibility in decision making. Although SFMs offer a strong representative framework for membership, nonmembership, and hesitancy under spherical constraints, they are restricted to fixed scalar values, which may not represent variability or impre- cision in real-world scenarios. IVSFM extends this to use interval values, hence giving a more comprehensive representation of uncertainty. The paper is organized as follows: In Section 2, some basic definitions are introduced, and these definitions help readers to understand the paper easily. In Section 3, we introduce the new concept of IVSFM with the basic definitions and discuss some important properties and theorems. In Section 4, we find determinants and adjoints for IVSFM with the help of an example, and some fundamental properties are examined. In Section 5, we introduce the EIVSFS and pro- vide algorithms for finding the GEIVSFS and LEIVSFS with the help of illustrations. In Section 6, we present a new distance measure for IVSFSs, investigate their properties, and explore their potential applications in decision making. The comparative study with existing work is conducted in Section 7. In Section 8, the conclusion of the paper is given with future directions. 2. Preliminaries Definition 1. [5] An Intuitionistic Fuzzy Set (IFS) A in universal set X is defined as a collection of the form A = {⟨x, µA(x), νA(x)⟩/x ∈ X}, where, µA : X → [0, 1] member ship function and νA : X → [0, 1] non-membership function of the element x ∈ X and ∀ x ∈ X such that 0 ≤ µA(x) + νA(x) ≤ 1. Definition 2. [17] Let X = {x1, x2, ...xm} represent a set of alternatives and Y = {y1, y2, ...yn} represents a set of attributes associated with each element of X. An in- tuitionistic fuzzy matrix is defined as A = (⟨(xi, yj), µA(xi, yj), νA(xi, yj)⟩) for i = 1, 2...m and j = 1, 2, ...n, where µA : X × Y → [0, 1] and νA : X × Y → [0, 1] satisfy the prop- erty 0 ≤ µA(xi, yj) + νA(xi, yj) ≤ 1. Intuitionistic fuzzy matrix denoted and defined as A = ( 〈 aij , a ′ ij 〉 ) such that aij + a′ij ≤ 1 for all i, j. Set of all intuitionistic fuzzy matrices of order of order m × n is denoted by Fmn and Fn denotes the set of intuitionistic fuzzy matrix, order n. Definition 3. [17] Pythagorean fuzzy matrix of size m×n is expressed as A = ( 〈 µaij , νaij 〉 ), µaij ∈ [0, 1] is membership value and νaij ∈ [0, 1] non-membership value of the ijth element with property: 0 ≤ µaij + νaij ≤ 1 for all i,j. Definition 4. [33] A picture fuzzy matrix A of the form, A = ( 〈 µaij , ηaij , νaij 〉 ), µaij , ηaij , νaij ∈ [0, 1] with the condition 0 ≤ µaij + ηaij + νaij ≤ 1 ∀ i,j. Where, µaij ∈ [0, 1] is the membership degree, ηaij ∈ [0, 1] is neutral membership degree, and νaij ∈ [0, 1] is non-membership degree. Definition 5. [38] A spherical fuzzy matrix A of the form, A = ( 〈 µaij , ηaij , νaij 〉 ) of a non negative real numbers µaij , ηaij , νaij ∈ [0, 1] satisfying the condition R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 5 of 24 0 ≤ µ2 aij+η2aij+ν2aij ≤ 1 for all i,j µaij ∈ [0, 1] is called the degree of membership,ηaij ∈ [0, 1] is called the degree of neutral membership and νaij ∈ [0, 1]is called the degree of non- membership Definition 6. [39] An Interval valued picture fuzzy matrix A is defined as A = (aij) = (⟨aijµ, aijη, aijυ⟩), i = 1, 2, ...m, j = 1, 2, ...n where aijµ = [aijµL, aijµU ] ∈ [0, 1], aijη = [aijηL, aijηU ] ∈ [0, 1] and aijυ = [aijυL, aijυU ] ∈ [0, 1] with the condition (aijµU ) + (aijηU ) + (aijυU ) ≤ 1 where aijµ, aijη and aijυ are the membership , neutral membership and non-membership degree of aij 3. Interval valued spherical fuzzy matrix We define an interval valued spherical fuzzy matrix (IVSFM) and its basic concepts by generalizing and extending the concept of spherical fuzzy matrix. Definition 7. An IVSFM A is expressed as A = (aij) = (⟨aijµ, aijη, aijυ⟩), i = 1, 2, ...m, j = 1, 2, ...n where aijµ = [aijµL, aijµU ] ∈ [0, 1], aijη = [aijηL, aijηU ] ∈ [0, 1] and aijυ = [aijυL, aijυU ] ∈ [0, 1] with the condition (aijµU ) 2 + (aijηU ) 2 + (aijυU ) 2 ≤ 1 where, aijµ is membership, aijη is neutral membership and aijυ is non-membership degree. Definition 8. An IVSFM is called a square interval valued spherical fuzzy matrix (SIVSFM) if it has the same number of rows and number columns. Definition 9. Let A and B be two IVSFM, such that A = ([aijµL, aijµU ], [aijηL, aijηU ], [aijυL, aijυU ]) and B = ([bijµL, bijµU ], [bijηL, bijηU ], [bijυL, bijυU ]). Then, A ≤ B if aijµL ≤ bijµL, aijµU ≤ bijµU ; aijηL ≤ bijηL, aijηU ≤ bijηU ; aijυL ≥ bijυL, aijυU ≥ bijυU . Definition 10. An IVSFM A is said to be a null matrix if [aijµL, aijµU ] = [0, 1], [aijηL, aijηU ] = [0, 1] and [aijυL, aijυU ] = [0, 1] for all i = 1, 2, ...m j = 1, 2, ...n Definition 11. An IVSFM A is said to be µ universal if [µijL, µijU ] = [1, 0], [ηijL, ηijU ] = [0, 1] and [υijL, υijU ] = [0, 1] for all i = 1, 2, ...m j = 1, 2, ...n Definition 12. An IVSFM A is said to be η universal if [µijL, µijU ] = [0, 1], [ηijL, ηijU ] = [1, 0] and [υijL, υijU ] = [0, 1] for all i = 1, 2, ...m j = 1, 2, ...n Definition 13. An IVSFM A is said to be υ universal if [µijL, µijU ] = [0, 1], [ηijL, ηijU ] = [0, 1] and [υijL, υijU ] = [1, 0] for all i = 1, 2, ...m j = 1, 2, ...n R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 6 of 24 3.1. Some Basic operations of IVSFM Definition 14. Let A and B be two IVSFM, of order m× n, A = ([aijµL, aijµU ], [aijηL, aijηU ], [aijυL, aijυU ]) and B = ([bijµL, bijµU ], [bijηL, bijηU ], [bijυL, bijυU ]). Then, 1. Ac = ([aijυL, aijυU ], [aijηL, aijηU ], [aijµL, aijµU ]) 2. A ∨B = ([max(aijµL, bijµL),max(aijµU , bijµU )] [min(aijηL, bijηL)], [min(aijηU , bijηU )][min(aijυL, bijυL)][min(aijυU , bijυU )]) 3. A ∧B = ([min(aijµL, bijµL),min(aijµU , bijµU )] [min(aijηL, bijηL)], [min(aijηU , bijηU )][max(aijυL, bijυL)][max(aijυU , bijυU )]) 4. AT = ([ajiυL, ajiυU ], [ajiηL, ajiηU ], [ajiµL, ajiµU ]) 5. A⊕B = ([ √ aijµL2 + bijµL 2 − aijµL2bijµL 2, √ aijµU 2 + bijµU 2 − aijµU 2bijµU 2] [aijηLbijηL, aijηUbijηU ][aijυLbijυL, aijυUbijυU ]) 6. A⊗B = ([aijµLbijµL, aijµUbijµU ][ √ aijηL2 + bijηL 2 − aijηL2bijηL 2, √ aijηU 2 + bijηU 2 − aijηU 2bijηU 2] [ √ aijυL2 + bijυL 2 − aijυL2bijυL 2, √ aijυU 2 + bijυU 2 − aijυU 2bijυU 2]) 7. An = ([[(aijµL) n, (aijµU ) n][ √ 1− (1− (aijµL)2)n, √ 1− (1− (aijµU )2)n] [ √ 1− (1− (aijυL)2)n, √ 1− (1− (aijυU )2)n]]) Theorem 1. Suppose A = ([aijµL, aijµU ], [aijηL, aijηU ], [aijυL, aijυU ]), B = ([bijµL, bijµU ], [bijηL, bijηU ], [bijυL, bijυU ]) and C = ([cijµL, cijµU ], [cijηL, cijηU ], [cijυL, cijυU ]) are three IVSFM of same order order m×n then 1. A ∨B = B ∨A 2. A ∧B = B ∧A 3. (AT )T = A 4. (Ac)T 5. A ∨ (B ∧ C) = (A ∨B) ∧ (A ∨ C) 6. A ∧ (B ∨ C) = (A ∧B) ∨ (A ∧ C) 7. A⊕B = B ⊕A 8. A⊗B = B ⊗A 9. A⊕ (B ⊕ C) = (A⊕B)⊕ C 10. A⊗ (B ⊗ C) = (A⊗B)⊗ C 11. (a) A⊗ (B ⊕ C) ̸= A⊗B)⊕ (A⊗ C) (b) (B ⊕ C)⊗ C ̸= (B ⊗A)⊕ (C ⊗A) Proof. 1. Let A ∨B = ([max(aijµL, bijµL),max(aijµU , bijµU )][min(aijηL, bijηL)], [min(aijηU , bijηU )] [min(aijυL, bijυL)][min(aijυU , bijυU )]) = ([max(bijµL, aijµL),max(bijµU , aijµU )][min(bijηL, aijηL)], [min(bijηU , aijηU )] R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 7 of 24 [min(bijυL, aijυL)][min(bijυU , aijυU )]) = B ∨A 2. Let A ∧B = ([min(aijµL, bijµL),min(aijµU , bijµU )][min(aijηL, bijηL)], [min(aijηU , bijηU )] [max(aijυL, bijυL)][max(aijυU , bijυU )]) = ([min(bijµL, aijµL),min(bijµU , aijµU )][min(bijηL, aijηL)], [min(bijηU , aijηU )] [max(bijυL, aijυL)][max(bijυU , aijυU )]) = B ∨A 3. Let(AT ) = ([ajiυL, ajiυU ], [ajiηL, ajiηU ], [ajiµL, ajiµU ]) Now, (AT )T = ([ajiυL, ajiυU ], [ajiηL, ajiηU ], [ajiµL, ajiµU ]) T = ([aijυL, aijυU ], [aijηL, aijηU ], [aijµL, aijµU ]). The properties 4,5 and 6 are straightforward. 7. A⊕B = ([ √ aijµL2 + bijµL 2 − aijµL2bijµL 2, √ aijµU 2 + bijµU 2 − aijµU 2bijµU 2] [aijηLbijηL, aijηUbijηU ][aijυLbijυL, aijυUbijυU ]) = ([ √ bijµL 2 + aijµL2 − bijµL 2aijµL2, √ bijµU 2 + aijµU 2 − bijµU 2aijµU 2] [bijηLaijηL, bijηUaijηU ][bijυLaijυL, bijυUaijυU ]) = B ⊕A 8. A⊗B = ([aijµLbijµL, aijµUbijµU ] [ √ aijηL2 + bijηL 2 − aijηL2bijηL 2, √ aijηU 2 + bijηU 2 − aijηU 2bijηU 2] [ √ aijυL2 + bijυL 2 − aijυL2bijυL 2, √ aijυU 2 + bijυU 2 − aijυU 2bijυU 2]) = ([bijµLaijµL, bijµUaijµU ] [ √ bijηL 2 + aijηL2 − bijηL 2aijηL2, √ bijηU 2 + aijηU 2 − bijηU 2aijηU 2] [ √ bijυL 2 + aijυL2 − bijυL 2aijυL2, √ bijυU 2 + aijυU 2 − bijυU 2aijυU 2]) = B ⊗A 9. A⊕ (B ⊕ C) = ([aijµL, aijµU ], [aijηL, aijηU ], [aijυL, aijυU ])⊕ ([ √ bijµL 2 + cijµL2 − bijµL 2cijµL2, √ bijµU 2 + cijµU 2 − bijµU 2cijµU 2] [bijηLcijηL, bijηUcijηU ][bijυLcijυL, bijυUcijυU ]) = ([ √ aijµL2 + bijµL 2 − aijµL2bijµL 2, √ aijµU 2 + bijµU 2 − aijµU 2bijµU 2] [aijηLbijηL, aijηUbijηU ][aijυLbijυL, aijυUbijυU ])⊕ ([cijµL, cijµU ], [cijηL, cijηU ], [cijυL, cijυU ]) = (A⊕B)⊕ C 10. Similarly A⊗ (B ⊗ C) = (A⊗B)⊗ C can be proved. 11. (a) A⊗ (B ⊕ C) ̸= (A⊗B)⊕ (A⊗ C) Let (B ⊕ C) = ([ √ bijµL 2 + cijµL2 − bijµL 2cijµL2, √ bijµU 2 + cijµU 2 − bijµU 2cijµU 2] [bijηLcijηL, bijηUcijηU ][bijυLcijυL, bijυUcijυU ]) Now, A⊗ (B ⊕ C) = ([aijµL( √ bijµL 2 + cijµL2 − bijµL 2cijµL2), aijµU ( √ bijµU 2 + cijµU 2 − bijµU 2cijµU 2)] R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 8 of 24 [ √ aijηL2 + bijηLcijηL 2 − aijηL2bijηLcijηL 2, √ aijηU 2 + bijηUcijηU 2 − aijηU 2bijηUcijηU 2] [ √ aijυL2 + bijηLcijηL 2 − aijυL2bijυLcijυL 2, √ aijυU 2 + bijυUcijυU 2 − aijυU 2bijυUcijυU 2]) A⊗B = ([aijµLbijµL, aijµUbijµU ][ √ aijηL2 + bijηL 2 − aijηL2bijηL 2, √ aijηU 2 + bijηU 2 − aijηU 2bijηU 2] [ √ aijυL2 + bijυL 2 − aijυL2bijυL 2, √ aijυU 2 + bijυU 2 − aijυU 2bijυU 2]) A⊗ C = ([aijµLcijµL, aijµUcijµU ][ √ aijηL2 + cijηL2 − aijηL2cijηL2, √ aijηU 2 + cijηU 2 − aijηU 2cijηU 2] [ √ aijυL2 + cijυL2 − aijυL2cijυL2, √ aijυU 2 + cijυU 2 − aijυU 2cijυU 2]) (A⊗B)⊕ (A⊗ C) = ([aijµL √ bijµL 2 + cijµL2 − bijµL 2cijµL2, aijµU √ bijµU 2 + cijµU 2 − bijµU 2cijµU 2]) [( √ aijηL2 + bijηL 2 − aijηL2bijηL 2)( √ aijηL2 + cijηL2 − aijηL2cijηL2), ( √ aijηU 2 + bijηU 2 − aijηU 2bijηU 2)( √ aijηU 2 + cijη2 − aijηU 2cijηU 2)], [( √ aijυL2 + bijυL 2 − aijυL2bijυL 2)( √ aijυL2 + cijυL2 − aijυL2cijυL2), ( √ aijυU 2 + bijυU 2 − aijυU 2bijυU 2)( √ aijυU 2 + cijυ2 − aijυU 2cijυU 2)] So, A⊗ (B ⊕ C) ̸= (A⊗B)⊕ (A⊗ C) (b) Similar manner, we can prove (B ⊕ C)⊗ C ̸= (B ⊗A)⊕ (C ⊗A) Theorem 2. Let A and B be IVSFM of size n, then (1) (A ∨B)c = Ac ∧Bc (2) (A ∧B)c = Ac ∨Bc Proof. (1) Let A = ([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), B = ([(bijµL, bijµU )], [(bijηL, bijηU )], [(bijυL, bijυU )]) Then, Ac = ([(aijυL, aijυU )], [(aijηL, aijηL), (aijµL, aijµL)], [(aijµL, aijµU )]), Bc = ([(bijυL, bijυU )], [(bijηL, bijηL), (bijµL, bijµL)], [(bijµL, bijµU )]), Let Ac ∧Bc = ([min(aijυL, bijυL),min(aijυU , bijυU )][min(aijηL, bijηL),min(aijηU , bijηU )] [max(aijµL, bijµL),max(aijµU , bijµU )]) A ∧B = ([max(aijυL, bijυL),max(aijυU , bijυU )][min(aijηL, bijηL),min(aijηU , bijηU )] [min(aijµL, bijµL),min(aijµU , bijµU )]) then, (A ∧B)c = ([min(aijυL, bijυL),min(aijυU , bijυU )][min(aijηL, bijηL),min(aijηU , bijηU )] [max(aijµL, bijµL),max(aijµU , bijµU )]) (2) Can be proved similarly. Theorem 3. Let A, B and C be IVSFM of size m× n and A ≤ C and B ≤ C, then A ∨B ≤ C Proof. Let A = ([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), B = ([(bijµL, bijµU )], [(bijηL, bijηU )], [(bijυL, bijυU )]) C = ([(cijµL, cijµU )], [(cijηL, cijηU )], [(cijυL, cijυU )]) R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 9 of 24 If A ≤ C then aijµL ≤ cijµL, aijµU ≤ cijµU , aijηL ≤ CijηL, aijηU ≤ aijηU , aijυL ≥ cijυL, aijυU ≥ cijυU ∀ i,j, and B ≤ C then bijµL ≤ cijµL, bijµU ≤ cijµU , bijηL ≤ CijηL, bijηU ≤ bijηU , bijυL ≥ cijυL, bijυU ≥ cijυU ∀ i,j. Now , max(aijµL, bijµL) ≤ cijµL,max(aijµU , bijµU ) ≤ cijµU ,min(aijηL, bijηL) ≤ cijηL, min(aijηU , bijµU ) ≤ cijηU ,min(aijυL, bijυL) ≥ cijυL,min(aijυU , bijυU ) ≥ cijυU . Hence A ∨B ≤ C (by def.3.3). Theorem 4. Let A, B and C be IVSFM of size m× n and A ≤ B, then A ∨ C ≤ B ∨ C Proof. Let A = ([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), B = ([(bijµL, bijµU )], [(bijηL, bijηU )], [(bijυL, bijυU )]) C = ([(cijµL, cijµU )], [(cijηL, cijηU )], [(cijυL, cijυU )]). If A ≤ B then, aijµL ≤ bijµL, aijµU ≤ bijµU , aijηL ≤ bijηL, aijηU ≤ bijηU , aijυL ≥ bijυL, aijυU ≥ bijυU , now max(aijµL, CijµL) ≤ max(bijµL, cijµL),max(aijµU , cijµU ) ≤ max(bijµU , cijµU ),min(aijηL, cijηL) ≤ min(aijηL, cijηL),min(aijηU , cijµU ) ≤ min(bijηU , cijηU ),min(aijυL, cijυL) ≥ min(bijυL, cijυL),min(aijυU , cijυU ) ≥ min(bijυU , cijυU ) ∀ i,j. Hence A ∨ C ≤ B ∨ C. Theorem 5. Let A, B and C be IVSFM of size m × n and C ≤ A and C ≤ B, then C ≤ A ∧B Proof. Followed from Theorem 3.12. Theorem 6. Let A, B and C be IVSFM of size m × n and A ≤ B and A ≤ C and B ∧ C = 0 then, then A = 0 Proof. If A ≤ B then, aijµL ≤ bijµL, aijµU ≤ bijµU , aijηL ≤ bijηL, aijηU ≤ bijηU , aijυL ≥ bijυL, aijυU ≥ bijυU . A ≤ C then, aijµL ≤ cijµL, aijµU ≤ cijµU , aijηL ≤ cijηL, aijηU ≤ cijηU , aijυL ≥ cijυL, aijυU ≥ cijυU . Hence by theorem 3.13 A ≤ B ∧ C,B ∧ C = 0 such that A = 0. Theorem 7. Let A, B and C be IVSFM of size m× n and A ≤ B then A ≤ C ≤ B ∧ C. Proof. The proof follows from Definition 3.3. Theorem 8. Let A, B and C be IVSFM of size m × n and A ≤ B and B ≤ C = 0 and then A ∧ C = 0. Proof. The proof follows from Theorem 3.15. R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 10 of 24 4. Determinant and adjoint of interval valued spherical fuzzy matrix This section focus on the determinant and adjoint of the IVSFM, illustrated with the help of examples, and some fundamental properties are also examined. Definition 15. Determinant of IVSFM, assume A = ([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), is an IVSFM of size n. The determi- nant of A is denoted by |A| and expressed as |A| = (∨h∈Pk ([a1h(1)µL, a1h(1)µU ] ∧ [a2h(2)µL, a2h(2)µU ]... ∧ [akh(k)µL, akh(k)µU ]), ∧h∈Pk ([a1h(1)ηL, a1h(1)ηU ] ∧ [a2h(2)ηL, a2h(2)ηU ]... ∧ [akh(k)ηL, akh(k)ηU ]), ∧h∈Pk ([a1h(1)υL, a1h(1)υU ] ∧ [a2h(2)υL, a2h(2)υU ]... ∧ [akh(k)υL, akh(k)υU ]). Let, PK denote set of permutation of the set {1, 2, 3, ...n}. Example 1. Consider an IVSFM of size 3 |A| =[0.30, 0.70][0.40, 0.20][0.20, 0.30] [0.40, 0.60][0.45, 0.40][0.21, 0.30] [0.71, 0.20][0.62, 0.10][0.10, 0.80] [0.32, 0.02][0.72, 0.20][0.33, 0.40] [0.23, 0.60][0.40, 0.50][0.21, 0.10] [0.71, 0.20][0.62, 0.20][0.30, 0.70] [0.36, 0.60][0.53, 0.20][0.65, 0.30] [0.40, 0.20][0.25, 0.70][0.21, 0.50] [0.15, 0.20][0.62, 0.20][0.18, 0.40]  to calculate the determinant of A, we need to determine all the permutations of the {1, 2, 3}. The permutations on {1, 2, 3} are ϕ1 = I, ϕ2 = (23), ϕ3 = (12), ϕ4 = (13), ϕ5 = (123), ϕ6 = (132) The membership component of |A| = ([a1ϕ1(1)µL, a1ϕ1(1)µU ] ∧ [a2ϕ1(2)µL, a1ϕ1(2)µU ]∧ [a3ϕ1(3)µL, a3ϕ1(3)µU ]) ∨ ([a1ϕ2(1)µL, a1ϕ2(1)µU ] ∧ [a2ϕ2(2)µL, a1ϕ2(2)µU ]∧ [a3ϕ2(3)µL, a3ϕ2(3)µU ]) ∨ ([a1ϕ3(1)µL, a3ϕ3(1)µU ]∧ [a2ϕ3(2)µL, a1ϕ3(2)µU ] ∧ [a3ϕ3(3)µL, a3ϕ3(3)µU ]) ∨ ([a1ϕ4(1)µL, a1ϕ4(1)µU ] ∧[a2ϕ4(2)µL, a1ϕ4(2)µU ] ∧ [a3ϕ4(3)µL, a3ϕ4(3)µU ]) ∨ ([a1ϕ5(1)µL, a1ϕ5(1)µU ] ∧[a2ϕ5(2)µL, a1ϕ5(2)µU ] ∧ [a3ϕ5(3)µL, a3ϕ5(3)µU ]) ∨ ([a1ϕ6(1)µL, a1ϕ6(1)µU ]∧ [a2ϕ6(2)µL, a1ϕ6(2)µU ] ∧ [a3ϕ6(3)µL, a3ϕ6(3)µU ]) = ([a11µL, a11µU ] ∧ [a22µL, a22µU ][a33µL, a33µU ]) ∨ ([a11µL, a11µU ] ∧ [a23µL, a23µU ] ∧ [a32µL, a32µU ]) ∨([a12µL, a12µU ] ∧ [a21µL, a21µU ] ∧ [a33µL, a33µU ]) ∨ ([a12µL, a12µU ] ∧ [a23µL, a23µU ] ∧ [a31µL, a31µU ]) ∨ ([a13µL, a13µU ] ∧ [a21µL, a21µU ] ∧ [a32µL, a32µU ]) ∨ [a13µL, a13µU ] ∧ ([a22µL, a22µU ] ∧ [a31µL, a31µU ]) = ([0.30, 0.70]∧[0.23, 0.60]∧[0.15, 0.20])∨([0.30, 0.70]∧[0.71, 0.20]∧[0.40, 0.20])∨([0.40, 0.60]∧ [0.32, 0.02]∧[0.15, 0.20])∨([0.40, 0.60]∧[0.71, 0.20]∧[0.36, 0.60])∨([0.71, 0.20]∧[0.32, 0.02]∧ [0.40, 0.20]) ∨ ([0.71, 0.20] ∧ [0.23, 0.60] ∧ [0.36, 0.60]) = [0.15, 0.20] ∨ [0.30, 0.20] ∨ [0.15, 0.02] ∨ [0.36, 0.20] ∨ [0.32, 0.02] ∨ [0.23, 0.20] = [0.36, 20] The neutral component of |A| = ([a1ϕ1(1)ηL, a1ϕ1(1)ηU ] ∧ [a2ϕ1(2)ηL, a1ϕ1(2)ηU ] ∧ [a3ϕ1(3)ηL, a3ϕ1(3)ηU ])∨ ([a1ϕ2(1)ηL, a1ϕ2(1)ηU ] ∧ [a2ϕ2(2)ηL, a1ϕ2(2)ηU ] ∧ [a3ϕ2(3)ηL, a3ϕ2(3)ηU ])∨ ([a1ϕ3(1)ηL, a3ϕ3(1)ηU ] ∧ [a2ϕ3(2)ηL, a1ϕ3(2)ηU ] ∧ [a3ϕ3(3)ηL, a3ϕ3(3)ηU ])∨ ([a1ϕ4(1)ηL, a1ϕ4(1)ηU ] ∧ [a2ϕ4(2)ηL, a1ϕ4(2)ηU ] ∧ [a3ϕ4(3)ηL, a3ϕ4(3)ηU ])∨ R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 11 of 24 ([a1ϕ5(1)ηL, a1ϕ5(1)ηU ] ∧ [a2ϕ5(2)ηL, a1ϕ5(2)ηU ] ∧ [a3ϕ5(3)ηL, a3ϕ5(3)ηU ])∨ ([a1ϕ6(1)ηL, a1ϕ6(1)ηU ] ∧ [a2ϕ6(2)ηL, a1ϕ6(2)ηU ] ∧ [a3ϕ6(3)ηL, a3ϕ6(3)ηU ]) = ([a11ηL, a11ηU ]∧[a22ηL, a22ηU ][a33ηL, a33ηU ])∨([a11ηL, a11ηU ]∧[a23ηL, a23ηU ]∧[a32ηL, a32ηU ])∨ ([a12ηL, a12ηU ]∧[a21ηL, a21ηU ]∧[a33ηL, a33ηU ])∨([a12ηL, a12ηU ]∧[a23ηL, a23ηU ]∧[a31ηL, a31ηU ])∨ ([a13ηL, a13ηU ] ∧ [a21ηL, a21ηU ] ∧ [a32ηL, a32ηU ]) ∨ ([a13ηL, a13ηU ] ∧ [a22µL, a22ηU ] ∧ [a31ηL, a31ηU ]) = ([0.40, 0.20]∧[0.40, 0.50]∧[0.62, 0.20])∨([0.40, 0.20]∧[0.62, 0.20]∧[0.62, 0.20])∨([0.45, 0.40]∧ [0.72, 0.20]∧[0.62, 0.20])∨([0.45, 0.40]∧[0.62, 0.20]∧[0.53, 0.20])∨([0.62, 0.10]∧[0.72, 0.20]∧ [0.25, 0.70]) ∨ ([0.62, 0.10] ∧ [0.40, 0.50] ∧ [0.53, 0.20) = [0.40, 0.20] ∨ [0.40, 0.20] ∨ [0.45, 0.20] ∨ [0.45, 0.20] ∨ [0.25, 0.10] ∨ [0.40, 0.10] = [0.45, 20] The non membership component of |A| = ([a1ϕ1(1)υL, a1ϕ1(1)υU ] ∧ [a2ϕ1(2)υL, a1ϕ1(2)υU ] ∧ [a3ϕ1(3)υL, a3ϕ1(3)υU ])∨ ([a1ϕ2(1)υL, a1ϕ2(1)υU ] ∧ [a2ϕ2(2)υL, a1ϕ2(2)υU ] ∧ [a3ϕ2(3)υL, a3ϕ2(3)υU ])∨ ([a1ϕ3(1)υL, a3ϕ3(1)υU ] ∧ [a2ϕ3(2)υL, a1ϕ3(2)υU ] ∧ [a3ϕ3(3)υL, a3ϕ3(3)υU ])∨ ([a1ϕ4(1)υL, a1ϕ4(1)υU ] ∧ [a2ϕ4(2)υL, a1ϕ4(2)υU ] ∧ [a3ϕ4(3)υL, a3ϕ4(3)υU ])∨ ([a1ϕ5(1)υL, a1ϕ5(1)υU ] ∧ [a2ϕ5(2)υL, a1ϕ5(2)υU ] ∧ [a3ϕ5(3)υL, a3ϕ5(3)υU ])∨ ([a1ϕ6(1)υL, a1ϕ6(1)υU ] ∧ [a2ϕ6(2)υL, a1ϕ6(2)υU ] ∧ [a3ϕ6(3)υL, a3ϕ6(3)υU ]) = ([a11υL, a11υU ] ∧ [a22υL, a22υU ][a33υL, a33υU ]) ∨ ([a11υL, a11υU ] ∧ [a23υL, a23υU ] ∧ [a32υL, a32υU ]) ∨ ([a12υL, a12υU ] ∧ [a21υL, a21υU ] ∧ [a33υL, a33υU ]) ∨ ([a12υL, a12υU ] ∧ [a23υL, a23υU ] ∧ [a31υL, a31υU ]) ∨ ([a13υL, a13υU ] ∧ [a21υL, a21υU ] ∧ [a32υL, a32υU ]) ∨ ([a13υL, a13υU ] ∧ [a22υL, a22υU ] ∧ [a31υL, a31υU ]) = ([0.20, 0.30]∧[0.21, 0.10]∧[0.18, 0.40)∨([0.20, 0.30]∧[0.30, 0.70]∧[0.21, 0.50])∨([0.21, 0.30]∧ [0.33, 0.40]∧ [0.18, 0.40)∨([0.21, 0.30]∧ [0.30, 0.70]∧ [0.65, 0.30])∨([0.10, 0.80]∧ [0.33, 0.40]∧ [0.21, 0.50]) ∨ ([0.10, 0.80] ∧ [0.21, 0.10] ∧ [0.65, 0.30]) = [0.18, 0.10] ∨ [0.20, 0.30] ∨ [0.18, 0.30] ∨ [0.21, 0.30] ∨ [0.10, 0.40] ∨ [0.10, 0.10] = [0.21, 40] Definition 16. Adjoint of IVSFM, Let A = ([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), be IVSFM of size n. So, the adjoint of A is denoted by adj([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]), and defined as adj([(aijµL, aijµU )], [(aijηL, aijηU )], [(aijυL, aijυU )]) = S = ([sijµ, sijη, sijυ]) = ([(sijµL, sijµU )], [(sijηL, sijηU )], [(sijυL, sijυU )]) = (⟨sijµ, sijη, sijυ⟩) where sijµ = ∨δ∈Qmjmi ∧u∈mj auδ(u)µ sijη = ∧δ∈Qmjmi ∧u∈mj auδ(u)η sijυ = ∧δ∈Qmjmi ∧u∈mj auδ(u)υ Example 2. Consider an IVSFM of size 3. A =[0.30, 0.70][0.40, 0.20][0.20, 0.30] [0.40, 0.60][0.45, 0.40][0.21, 0.30] [0.71, 0.20][0.62, 0.10][0.10, 0.80] [0.32, 0.02][0.72, 0.20][0.33, 0.40] [0.23, 0.60][0.40, 0.50][0.21, 0.10] [0.71, 0.20][0.62, 0.20][0.30, 0.70] [0.36, 0.60][0.53, 0.20][0.65, 0.30] [0.40, 0.20][0.25, 0.70][0.21, 0.50] [0.15, 0.20][0.62, 0.20][0.18, 0.40]  Let j = 1 and i = 1, we have mj = {1, 2, 3}−{1} = {2, 3} and mi = {1, 2, 3}−{1} = {2, 3}. The permutation of mi over mj are ( 2 3 2 3 ) ( 2 3 3 2 ) Now, R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 12 of 24 (a22µ ∧ a33µ) ∨ (a23µ ∧ a32µ = ([0.23, 0.60] ∧ [0.15, 0.20]) ∨ ([0.71, 0.20] ∧ [0.40, 0.20]) = [0.15, 0.20] ∨ [0.40, 0.20] = [0.40, 0.20] (a22η ∧ a33η) ∨ (a23η ∧ a32η = ([0.40, 0.50] ∧ [0.62, 0.20]) ∨ ([0.62, 0.20] ∧ [0.25, 0.70]) = [0.40, 0.20] ∨ [0.25, 0.20] = [0.40, 0.20] (a22υ ∧ a33υ) ∨ (a23υ ∧ a32υ = ([0.21, 0.10] ∧ [0.18, 0.40]) ∨ ([0.30, 0.70] ∧ [0.21, 0.50]) = [0.18, 0.10] ∨ [0.21, 0.50] = [0.21, 0.50] For j = 1 and i = 2, mj = {1, 2, 3} − {1} = {2, 3} and mi = {1, 2, 3} − {2} = {1, 3}. The permutation of mi over mj are( 1 3 2 3 ) ( 1 3 3 2 ) Now, (a12µ ∧ a33µ) ∨ (a13µ ∧ a32µ = ([0.40, 0.60] ∧ [0.15, 0.20]) ∨ [0.71, 0.20] ∧ [0.40, 0.20]) = [0.15, 0.20] ∨ [0.40, 0.20] = [0.40, 0.20] (a12η ∧ a33η) ∨ (a13η ∧ a32η = ([0.45, 0.40] ∧ [0.62, 0.20]) ∨ ([0.62, 0.10] ∧ [0.25, 0.70]) = [0.45, 0.20] ∨ [0.25, 0.10] = [0.45, 0.20] (a12υ ∧ a33υ) ∨ (a13υ ∧ a32υ = ([0.21, 0.30] ∧ [0.18, 0.40]) ∨ ([0.30, 0.70] ∧ [0.10, 0.80]) = [0.18, 0.30] ∨ [0.10, 0.70] = [0.18, 0.70] For j = 1 and i = 3, mj = {1, 2, 3} − {1} = {2, 3} and mi = {1, 2, 3} − {3} = {1, 3}. The permutation of mi over mj are( 1 2 2 3 ) ( 1 2 3 2 ) Now, (a12µ ∧ a23µ) ∨ (a13µ ∧ a22µ = ([0.40, 0.60] ∧ [0.71, 0.20]) ∨ [0.71, 0.20] ∧ [0.23, 0.60]) = [0.40, 0.20] ∨ [0.23, 0.20] = [0.40, 0.20] (a12η ∧ a23η) ∨ (a13η ∧ a22η = ([0.45, 0.40] ∧ [0.62, 0.20]) ∨ ([0.62, 0.10] ∧ [0.40, 0.50]) = [0.45, 0.20] ∨ [0.40, 0.10] = [0.45, 0.20] (a12υ ∧ a23υ) ∨ (a13υ ∧ a22υ = ([0.21, 0.30] ∧ [0.30, 0.70]) ∨ ([0.30, 0.70] ∧ [0.21, 0.10]) = [0.21, 0.30] ∨ [0.21, 0.10] = [0.21, 0.30] Similar manner, we can find other values as well, Adjoint(A) is obtained as Adjoint(A)=[0.40, 0.20][0.40, 0.20][0.21, 0.50] [0.40, 0.20][0.40, 0.20][0.18, 0.70] [0.40, 0.20][0.40, 0.20][0.21, 0.30] [0.40, 0.20][0.62, 0.20][0.21, 0.50] [0.40, 0.20][0.40, 0.20][0.18, 0.50] [0.30, 0.20][0.40, 0.20][0.20, 0.30] [0.40, 0.20][0.25, 0.20][0.33, 0.50] [0.40, 0.20][0.25, 0.20][0.20, 0.50] [0.23, 0.60][0.40, 0.20][0.20, 0.10]  In the next section, we introduce eigen interval valued spherical fuzzy sets and propose the algorithms to determine least and greatest eigen interval valued spherical fuzzy sets. R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 13 of 24 5. Algorithms for eigen interval valued spherical fuzzy set In this section , we propose the EIVSFS and provide algorithms to determine the GEIVSFS and LEIVSFS with the help of illustration. Definition 17. An interval valued spherical fuzzy relation (IVSFR) between IVSFS X and Y expressed as: R = {((x, y).µR(x, y), ηR(x, y), υR(x, y)) : x ∈ X, y ∈ Y }, where µR = [µL R, µ U R], ηR = [ηLR, η U R ], υR = [υLR, υ U R ], such that 0 ≤ (µU R) 2 + (ηUR) 2 + (υUR) 2 ≤ 1 ∀ (x, y) ∈ (X × Y ). Consider R1 ∈ (X × y) and R2 ∈ (Y × Z) be IVSFR. Max-Min Operation Let the max-min composition operator for two IVSFR R1 ∈ (X×y) and R2 ∈ (Y ×Z) are defined as R1 ◦R2 = {((xij , zij), µR1◦R2(xij , zij), ηR1◦R2(xij , zij), υR1◦R2(xij , zij)) : xij ∈ X, zij ∈ Z} , where, µR1◦R2(xij , zij) = [µL R1◦R2 (xij , zij), µ U R1◦R2 (xij , zij)] ηR1◦R2(xij , zij) = [ηLR1◦R2 (xij , zij), η U R1◦R2 (xij , zij)] and υR1◦R2(xij , zij) = [υLR1◦R2 (xij , zij), υ U R1◦R2 (xij , zij)] In addition, µL R1◦R2 (xij , zij) = maxy∈Y { minx∈X(µL R1 (xij , zij), µ L R2 (xij , zij)) } , µU R1◦R2 (xij , zij) µU R1◦R2 (xij , zij) = maxy∈Y { minx∈X(µU R1 (xij , zij), µ U R2 (xij , zij)) } , µL R1◦R2 (xij , zij) ηLR1◦R2 (xij , zij) = miny∈Y { minx∈X(ηLR1 (xij , zij), η L R2 (xij , zij)) } , ηUR1◦R2 (xij , zij) ηUR1◦R2 (xij , zij) = miny∈Y { minx∈X(ηUR1 (xij , zij), η U R2 (xij , zij)) } , ηLR1◦R2 (xij , zij) υLR1◦R2 (xij , zij) = miny∈Y { maxx∈X(υLR1 (xij , zij), υ L R2 (xij , zij)) } , υUR1◦R2 (xij , zij) υUR1◦R2 (xij , zij) = miny∈Y { maxx∈X(υUR1 (xij , zij), υ U R2 (xij , zij)) } , υLR1◦R2 (xij , zij) Min-Max Operation Let the min-max composition operator for two IVSFR be defined as R1 •R2 = {((xij , zij), µR1•R2(xij , zij), ηR1•R2(xij , zij), υR1•R2(xij , zij)) : xij ∈ X, zij ∈ Z} , where, µR1•R2(xij , zij) = [µL R1•R2 (xij , zij), µ U R1•R2 (xij , zij)] ηR1•R2(xij , zij) = [ηLR1•R2 (xij , zij), η U R1•R2 (xij , zij)] and υR1•R2(xij , zij) = [υLR1•R2 (xij , zij), υ U R1•R2 (xij , zij)] In addition, µL R1•R2 (xij , zij) = miny∈Y { maxx∈X(µL R1 (xij , zij), µ L R2 (xij , zij)) } , µU R1•R2 (xij , zij) µU R1•R2 (xij , zij) = miny∈Y { maxx∈X(µU R1 (xij , zij), µ U R2 (xij , zij)) } , µL R1•R2 (xij , zij) ηLR1•R2 (xij , zij) = miny∈Y { minx∈X(ηLR1 (xij , zij), η L R2 (xij , zij)) } , ηUR1•R2 (xij , zij) ηUR1•R2 (xij , zij) = miny∈Y { minx∈X(ηUR1 (xij , zij), η U R2 (xij , zij)) } , ηLR1•R2 (xij , zij) υLR1•R2 (xij , zij) = maxy∈Y { minx∈X(υLR1 (xij , zij), υ L R2 (xij , zij)) } , υUR1•R2 (xij , zij) υUR1•R2 (xij , zij) = maxy∈Y { minx∈X(υUR1 (xij , zij), υ U R2 (xij , zij)) } , υLR1•R2 (xij , zij) R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 14 of 24 Definition 18. Let R be an IVSFR defined on IVSFS of X. An IVSFS N is called an eigen interval valued spherical fuzzy set associated with the given relation R if N ∗R = N , ∗ is either the min-max or max-min operator. 5.1. Greatest Eigen interval valued spherical fuzzy set For finding GEIVSFS with the IVSFR R using the max-min composition operator. Let N1 represent the IVSFS, where the degree of membership is highest and the degrees of neutral membership and degree of non-membership is the lowest of all elements of the column of R: µN1(u) = maxx∈XµR(x, u)∀u ∈ Y, ηN1(u) = minx∈XηR(x, u)∀u ∈ y, υN1(u) = minx∈XυR(x, u)∀u ∈ y, (1) It is simple to check that N1 is an eigen inter-valued spherical fuzzy set, but not always be the GEIVSFS. To find GEIVSFS, the following steps are evaluated using max- min operation N1 ◦R = N2, N2 ◦R = N1 ◦R2 = N3, N3 ◦R = N1 ◦R3 = N4, ... Nn ◦R = N1 ◦Rn = Nn+1, Now, we provide an algorithm to calculate GEIVSFS. Algorithm A Step 1: Find the set N1 from R by using equation 1 Step 2: Choose the index n = 1 and calculate Nn+1 = Nn ◦R. Step 3: If Nn+1 ̸= Nn the go to step 2. Step 4: If Nn+1 = Nn then Nn is the GEIVSFS. Example 3. A =[0.30, 0.70][0.40, 0.20][0.20, 0.30] [0.40, 0.60][0.45, 0.40][0.21, 0.30] [0.71, 0.20][0.62, 0.10][0.10, 0.80] [0.32, 0.02][0.72, 0.20][0.33, 0.40] [0.23, 0.60][0.40, 0.50][0.21, 0.10] [0.71, 0.20][0.62, 0.20][0.30, 0.70] [0.36, 0.60][0.53, 0.20][0.65, 0.30] [0.40, 0.20][0.25, 0.70][0.21, 0.50] [0.15, 0.20][0.62, 0.20][0.18, 0.40]  Step 1: N1 = ([0.36, 0.70][0.40, 0.20][0.20, 0.30]) ([0.40, 0.60][0.25, 0.40][0.21, 0.10]) ([0.71, 0.20][0.62, 0.10][0.10, 0.40]) Step 2: Setting n = 1, N2 = N1 ◦R N2 = ([0.36, 0.70][0.25, 0.10][0.20, 0.30]) ([0.40, 0.60][0.25, 0.40][0.21, 0.10]) ([0.40, 0.20][0.25, 0.10][0.18, 0.40]) Step 3: N2 ̸= N1 then choose n=2 in step 2 and find N3 that is N3 = N2 ◦R: R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 15 of 24 Figure 1: Flow chart For Algorithm A N3 = ([0.36, 0.70][0.25, 0.10][0.20, 0.30]) ([0.40, 0.60][0.25, 0.10][0.21, 0.10]) ([0.40, 0.20][0.25, 0.10][0.18, 0.40]) Step 4: Since N3 = N2, is the GEIVSFS associated with R. 5.2. Least Eigen interval valued spherical fuzzy set For finding LEIVSFS with the IVSFR R using the max-min composition operator. Let N1 be the IVSFS, in which the degree of membership and the degree of neutral membership are the lowest of all the members of the column of R and the degree of non-membership is the highest of all the members of the column of R. µN1(u) = minx∈XµR(x, u)∀u ∈ Y, ηN1(u) = minx∈XηR(x, u)∀u ∈ y, υN1(u) = maxx∈XυR(x, u)∀u ∈ y, (2) It is easy to check that N1 is an eigen interval valued spherical fuzzy set, but it should not always be the LEIVSFS. To find LEIVSFS, the following steps are evaluated using max- min operation N1 ◦R = N2, N2 ◦R = N1 ◦R2 = N3, N3 ◦R = N1 ◦R3 = N4, ... Nn ◦R = N1 ◦Rn = Nn+1, R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 16 of 24 Now, we give an algorithm to find LEIVSFS. Algorithm B Step 1: Obtain the set N1 from R by equation 2 Step 2: Choosing the index n = 1 and calculating Nn+1 = Nn ◦R. Step 3: If Nn+1 ̸= Nn, go to Step 2. Step 4: If Nn+1 = Nn, then Nn is the LEIVSFS. Figure 2: Flow chart for Algorith B Example 4. A = [0.30, 0.70][0.40, 0.20][0.20, 0.30] [0.40, 0.60][0.45, 0.40][0.21, 0.30] [0.71, 0.20][0.62, 0.10][0.10, 0.80] [0.32, 0.02][0.72, 0.20][0.33, 0.40] [0.23, 0.60][0.40, 0.50][0.21, 0.10] [0.71, 0.20][0.62, 0.20][0.30, 0.70] [0.36, 0.60][0.53, 0.20+][0.65, 0.30] [0.40, 0.20][0.25, 0.70][0.21, 0.50] [0.15, 0.20][0.62, 0.20][0.18, 0.40]  Step 1: N1 = ([0.30,0.02][0.40,0.20][0.65,0.40]) ([0.23,0.20][0.25,0.40][0.21,0.50]) ([0.15,0.20][0.62,0.10][0.30,0.80]) Step 2: Set n = 1, N2 = N1 ◦R N2 = ([0.30, 0.20][0.25, 0.10][0.30, 0.40]) ([0.23, 0.20][0.25, 0.10][0.21, 0.50]) ([0.15, 0.20][0.25, 0.10][0.21, 0.50]) Step 3: N2 ̸= N1 then choose n = 2 in step 2 and find N3 that is N3 = N2 ◦R: R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 17 of 24 N3 = ([0.30, 0.20][0.25, 0.10][0.30, 0.40]) ([0.23, 0.20][0.25, 0.10][0.21, 0.50]) ([0.15, 0.20][0.25, 0.10][0.21, 0.50]) Hence, N3 = N2,so N3 is the LEIVSFS associated with R. 6. Distance measure in decision-making problem This section presents a distance measure for IVSFS, investigates its properties, and explores its potential applications in decision making using IVSFM. 6.1. Distance measure of IVSFSs Definition 19. let consider two IVSFSs A = {[a1(xi), b1(xi)], [c1(xi), d1(xi)], [e1(xi), f1(xi)]} B = {[a2(xi), b2(xi)], [c2(xi), d2(xi)], [e2(xi), f2(xi)]} the distance measure between A and B is defined as follows: d(A,B) = 1 3(|a1(xi)− a2(xi)|+ |b1(xi)− b2(xi)|+ |c1(xi)− c2(xi)| +|d1(xi)− d2(xi)|+ |e1(xi)− e2(xi)|+ |f1(xi)− f2(xi)|) where i = 1, 2, 3...n Theorem 9. The distance measure between IVSFSs A and B is a function d : IV SFSs → IV SFSs, which satisfies the following conditions 1. 0 ≤ d(A,B) ≤ 1 2. d(A,B) = 0 iff A = B 3. d(A,B) = d(B,A) (d1) 4. Let A,B,B ∈ IV SFSs then d(A,C) ≤ d(A,B) + d(B,C) Proof. 1. It is obvious d(A,B) ∈ [0, 1] 2. As a1 = a2, b1 = b2, c1 = c2, d1 = d2, e1 = e2, f1 = f2 such that d(A,B) = 0 iff if A = B 3. d(A,B) = 1 3(|a1(xi)− a2(xi)|+ |b1(xi)− b2(xi)|+ |c1(xi)− c2(xi)| +|d1(xi)− d2(xi)|+ |e1(xi)− e2(xi)|+ |f1(xi)− f2(xi)|) = 1 3( ∑ |a2(xi)− a1(xi)|+ |b2(xi)− b1(xi)|+ |c2(xi)− c1(xi)| +|d2(xi)− d1(xi)|+ |e2(xi)− e1(xi)|+ |f2(xi)− f1(xi)|) = d(B,A) 4. A = {[a1(xi), b1(xi)], [c1(xi), d1(xi)], [e1(xi), f1(xi)]} B = {[a2(xi), b2(xi)], [c2(xi), d2(xi)], [e2(xi), f2(xi)]} C = {[a3(xi), b3(xi)], [c3(xi), d3(xi)], [e3(xi), f3(xi)]} Consider d(A,C) = 1 3(|a1(xi)− a3(xi)|+ |b1(xi)− b3(xi)|+ |c1(xi)− c3(xi)| +|d1(xi)− d3(xi)|+ |e1(xi)− e3(xi)|+ |f1(xi)− f3(xi)|) = 1 3(|a1(xi)−a2(xi)+a2(xi)−a3(xi)|+ |b1(xi)−b2(xi)+b2(xi)−b3(xi)|+ |c1(xi)−c2(xi)+ c2(xi)− c3(xi)| +|d1(xi)− d2(xi)+ d2(xi)− d3(xi)|+ |e1(xi)− e2(xi)+ e2(xi)− e3(xi)|+ |f1(xi)− f2(xi)+ f2(xi)− f3(xi)|) ≤ 1 3(|a1(xi)− a2(xi)|+ |b1(xi)− b2(xi)|+ |c1(xi)− c2(xi)| R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 18 of 24 +|d1(xi)− d2(xi)|+ |e1(xi)− e2(xi)|+ |f1(xi)− f2(xi)|) +1 3(|a2(xi)− a3(xi)|+ |b2(xi)− b3(xi)|+ |c2(xi)− c3(xi)| +|d2(xi)− d3(xi)|+ |e2(xi)− e3(xi)|+ |f2(xi)− f3(xi)|) Hence d(A,C) ≤ d(A,B) + d(B,C) 6.2. Application in decision making problem To tackle the decision making problem of selecting administrative officers (AOs) for the promotion of Govt secretaries, we use the IVSFM. This approach is useful when dealing with uncertainty. Let us consider three Govts. G1, G2, G3 based on three political parties PP1, PP2 and PP3 . Five AOs, A1, A2, A3, A4 and A5 are shortlisted for promotion to the position of Govt Secretary. Consider IVSFM A of dimension 5× 3 which presents the view of An AO towards a Govt. supported by a political party. Consider the other IVSFM B of order 3 × 3 which shows the work carried by Govt. followed by a political party during the election period. Step A. In the IVSFM A, the information of each AOs is provided in relation with set of political parties {PP1, PP2, PP3}, while as in the IVSFM B, the information of each Govts. is given with respect to the same set of political parties {PP1, PP2, PP3}. On the basis of that concept, eight IVSFSs are obtained over the set {PP1, PP2, PP3} as follows: A1 = (PP1[0.40, 0.20][0.40, 0.20][0.21, 0.50]), (PP2[0.40, 0.20][0.40, 0.20][0.18, 0.70]), (PP3[0.40, 0.20][0.40, 0.10][0.21, 0.30]) A2 = (PP1[0.30, 0.20][0.62, 0.20][0.21, 0.50])(PP2[0.70, 0.20][0.50, 0.20][0.18, 0.50]), (PP3[0.30, 0.20][0.20, 0.20][0.60, 0.30]) A3 = (PP1[0.60, 0.20][0.25, 0.20][0.33, 0.50])(PP2[0.20, 0.20][0.25, 0.20][0.20, 0.50]), (PP3[0.23, 0.60][0.10, 0.20][0.20, 0.10]) A4 = (PP1[0.60, 0.20][0.25, 0.20][0.33, 0.50])(PP2[0.20, 0.20][0.25, 0.20][0.20, 0.50]), (PP3[0.23, 0.60][0.40, 0.20][0.30, 0.10]) A5 = (PP1[0.60, 0.20][0.25, 0.20][0.33, 0.50])(PP2[0.20, 0.20][0.25, 0.20][0.20, 0.50]), R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 19 of 24 (PP3[0.23, 0.60][0.30, 0.10][0.25, 0.70]) G1 = (PP1[0.20, 0.10][0.40, 0.10][0.30, 0.50])(PP2[0.40, 0.20][0.30, 0.20][0.18, 0.70]), (PP3[0.20, 0.20][0.30, 0.10][0.21, 0.30]) G2 = (PP1[0.20, 0.30][0.62, 0.20][0.20, 0.50])(PP2[0.60, 0.20][0.50, 0.20][0.18, 0.50]), (PP3[0.30, 0.20][0.20, 0.20][0.60, 0.20]) G3 = (PP1[0.60, 0.20][0.25, 0.20][0.33, 0.50])(PP2[0.20, 0.20][0.25, 0.20][0.20, 0.50]), (PP3[0.23, 0.60][0.10, 0.20][0.40, 0.20]) Step B. Using the distance measure between two IVSFSs the following distance matrix D of dimensions 5 × 3, is derived and dim represents the distance between Ai and Gm for i = 1, 2, 3, 4, 5 and m = 1, 2, 3 3. The following observation are made using the distance measure. 1. In Case of the Govt. G1, degree of closeness DOC between the AO A1 and the perfor- mance of Govt. G1 is maximum becauseDOC(A1, G1) > DOC(A2, G1) > DOC(A5, G1) > DOC(A4, G1) > DOC(A3, G1) 2. In Case of the Govt. G2, degree of closeness DOC between the AO A2 and the perfor- mance of Govt. G2 is maximum because DOC(A2, G2) > DOC(A1, G2) > DOC(A4, G2) > DOC(A3, G2) > DOC(A5, G2) 3. In case of the Govt. G3, degree of closeness (DOC) between the AO A3 and the perfor- mance of Govt. G3 is maximum becauseDOC(A3, G3) > DOC(A4, G3) > DOC(A1, G3) > DOC(A2, G3) > DOC(A5, G3) where > denotes the degree of closeness, the lower the value indicates a higher degree of closeness. As per calculation , the final selected list of AOs for different governments is determined as follows: Govt: AOs (G1) A1, A2 (G2) A2, A1 (G3) A3, A4 AOs A1, A2 are selected for all Govts. Fig. 3 illustrates a flow chart that states the procedure to be followed for AO selection. R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 20 of 24 Figure 3: Selection Procedure for AOs 7. Comparative analysis Whereas in previous studies on picture-fuzzy decision making, the information was accepted in picture-fuzzy form. But if we are to address various types of uncertainty in the information, conventional methods are not in a position to address such a situation. In such circumstances, we have to gather or represent the information in an interval-valued spherical fuzzy sense. In such instances, the process formulated so far becomes crucial in arriving at an effective and useful conclusion. Dogra and Pal [33] suggested a model for the selection of a selected group of adminis- trative officers for various governments on the basis of the distance measure between two picture fuzzy matrix. Their approach considered membership, neutral membership, and non-membership degrees in the picture fuzzy matrix context. In contrast, current research generalizes this work by considering membership, neutral membership, and nonmember- ship degrees as interval numbers, which has more relevance to real-world issues in the selection of administrative officers using the new distance measure. In addition, Ejegwa et al.[26] developed a model for computing students’ career paths based on the distance formula between two intuitionistic fuzzy sets. In this context, intu- itionistic fuzzy sets considered only membership and non-membership degrees. Moreover, Khalaf [27] solved medical diagnosis problems using the interval valued intuitionistic fuzzy set with max–min and min-max composition. They expressed these problems as uncertain decision matrices and provided decisions based on fuzzy scores for each attribute. However, our current method is distinct with matrices with interval valued spherical fuzzy values. We obtain interval valued spherical fuzzy sets from these matrices on a given uni- verse. Using the new formula for the distance between two IVSFSs, we obtain a distance matrix which leads to a decision. The use of this method is incredibly straightforward R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 21 of 24 since it does not include various complex calculations, hence avoiding any complication in its application. Therefore, developing an algorithm and computer programming for this method is straightforward. Besides, the data points utilized in this method have an exceptional capability to handle a greater level of vagueness in information. Recalling that the interval-valued spherical fuzzy concept is a generalization of the picture fuzzy concept, this study can be viewed as a generalization of higher order fuzzy logic. In summary, the study of IVSFM has immense advantages in the solution of real-world problems, particularly in the selection of administrative officers, manufacturing companies, and health insurance providers. These concepts are practical tools and subjects of research for various applications and, as such, are significant aspects in modern-day research and real-world problem solving scenarios. 8. Conclusion In this paper, we have introduced the concept of IVSFM along with some definitions and theorems. We proposed the definition of determinant and adjoint of IVSFM along with some related results. Further, we propose a formal definition of an EIVSFS for the interval- valued spherical fuzzy relations, along with the algorithms to find GEIVSFS and LEIVSFS by using min-max and max-min composition operators, respectively. Numerical examples have also been presented to illustrate the algorithms. In addition, the implementation of the GEIVPFS and LEIVSFS in decision-making problems has been demonstrated quite effectively. Furthermore, the new distance measure is proposed to solve the problems of decision making quite effectively with the help of IVSFM. Interval-valued spherical fuzzy matrices offer enhanced uncertainty modeling for multi-criteria decision-making, enabling more accurate and flexible assessments. They are highly applicable in real-world scenarios like medical diagnosis, supply chain management, and AI-based decision support systems where ambiguity and expert variability are critical. This study provides a platform for researchers to expand and generalize our findings across various types of data sets. It can be extended in fields such as image information retrieval, genetic algorithms for image reconstruction, and the concept of interval-valued eigen spherical fuzzy soft sets or soft matrices, which have been briefly stated for future research. Acknowledgements The authors would like to thank the Editor and reviewer’s for their valuable suggestions and comments to improve this article in the present form. References [1] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965. [2] R. H. Kim and F. W. Roush. Generalized fuzzy matrices. Fuzzy Sets and Systems, 4(3):293–315, 1980. R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 22 of 24 [3] M. G. Thomason. Convergence of powers of a fuzzy matrix. Journal of Mathematical Analysis and Applications, 57(2):476–480, 1977. [4] H. Hashimoto. Convergence of powers of a fuzzy transitive matrix. Fuzzy Sets and Systems, 9(1-3):153–160, 1983. [5] K. Atanassov. Intuitionistic fuzzy sets. International Journal of Bioautomation, 20(S1):1–6, 2016. [6] R. Verma. Generalized Bonferroni mean operator for fuzzy number intuitionistic fuzzy sets and its application to multi-attribute decision making. International Journal of Intelligent Systems, 30(5):499–519, 2015. [7] D. Xie, F. Xiao, and W. Pedrycz. Information quality for intuitionistic fuzzy val- ues with its application in decision-making. Engineering Applications of Artificial Intelligence, 109:104568, 2022. [8] W. Y. Zeng, H. H. S. Cui, Y. Q. Liu, Q. Yin, and Z. S. Xu. Novel distance measure between intuitionistic fuzzy sets and its application in pattern recognition. Iranian Journal of Fuzzy Systems, 19(3):127–137, 2022. [9] M. Luo and R. Zhao. A distance measure between intuitionistic fuzzy sets and its application in medical diagnosis. Artificial Intelligence in Medicine, 89:34–39, 2018. [10] N. X. Thao, M. Ali, and F. Smarandache. An intuitionistic fuzzy clustering algorithm based on a new correlation coefficient with application in medical diagnosis. Journal of Intelligent & Fuzzy Systems, 36(1):189–198, 2019. [11] S. El-Morsy. Stock portfolio optimization using Pythagorean fuzzy numbers. Journal of Operational and Strategic Analytics, 1(1):8–13, 2023. [12] K. Atanassov. Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31(3):343–349, 1989. [13] R. Verma and J. M. Merigó. A new decision-making method using interval-valued intuitionistic fuzzy cosine similarity measure based on the weighted reduced intuition- istic fuzzy sets. Informatica, 31(2):399–433, 2020. [14] R. Verma and S. Chandra. Interval-valued intuitionistic fuzzy-analytic hierarchy process for evaluating the impact of security attributes in fog-based internet of things paradigm. Computer Communications, 175:35–46, 2021. [15] Z. C. Liang, Y. Yang, and S. G. Liao. An interval-valued intuitionistic fuzzy two- sided matching model considering the level of automation. Applied Soft Computing, 116:108252, 2022. [16] S. Perçin. Circular supplier selection using interval-valued intuitionistic fuzzy sets. Environment, Development and Sustainability, 24:5551–5581, 2022. [17] M. Pal and S. K. Khan. Some operations on intuitionistic fuzzy matrices. Proceedings of the International Conference on Analysis and Discrete Structures, pages 22–24, 2002. [18] M. Bhowmik and M. Pal. Some results on intuitionistic fuzzy matrices and intu- itionistic circulant fuzzy matrices. International Journal of Mathematical Sciences, 7(1-2):177–192, 2008. [19] R. Pradhan and M. Pal. Convergence of maxarithmetic mean-minarithmetic mean powers of intuitionistic fuzzy matrices. International Journal of Fuzzy Mathematical R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 23 of 24 Archive, 2:58–69, 2013. [20] R. A. Padder and P. Murugadas. Convergence of powers and canonical form of s- transitive intuitionistic fuzzy matrix. New Trends in Mathematical Sciences, 5(2):229– 236, 2017. [21] T. Muthuraj and K. Lalitha. Some new operations and its properties on intuitionistic fuzzy matrices. International Journal of Research and Analytical Reviews, 4(4):735– 741, 2017. [22] T. Muthuraj and K. Lalitha. Some algebraic structures on max-min and min-max composition over intuitionistic fuzzy matrices. Advances in Mathematics: Scientific Journal, 9:5683–5691, 2020. [23] R. A. Padder and P. Murugadas. Max-max operation on intuitionistic fuzzy matrix. Annals of Fuzzy Mathematics and Informatics, 12(6):757–766, 2016. [24] P. Murugadas and R. A. Padder. Reduction of an intuitionistic fuzzy rectangular matrix. Annamalai University Science Journal, 49:15–18, 2015. [25] R. A. Padder and P. Murugadas. Determinant theory for intuitionistic fuzzy matrices. Afrika Matematika, 30:943–955, 2019. [26] P. A. Ejegwa, B. B. Tyongi, S. C. Osuwa, and A. A. Atime. Career determination using intuitionistic fuzzy multisets approach. MAYFEB Journal of Mathematics, 1:1–10, 2017. [27] M. M. Khalaf. Medical diagnosis via interval-valued intuitionistic fuzzy sets. Annals of Fuzzy Mathematics and Informatics, 6(2):245–249, 2013. [28] B. C. Cuong and V. Kreinovich. Picture fuzzy sets. Journal of Computer Science and Cybernetics, 30(4):409–420, 2014. [29] A. H. Ganie. A picture fuzzy distance measure and its application to pattern recog- nition problems. Iranian Journal of Fuzzy Systems, 20(1):71–85, 2023. [30] S. A. Wani, T. R. Shah, W. Ramı́rez, and C. Cesarano. Exploring the properties of multivariable Hermite polynomials in relation to Apostol-type Frobenius–Genocchi polynomials. Georgian Mathematical Journal, 2024. [31] M. Zayed, S. A. Wani, M. Subzar, and M. Riyasat. Certain families of differential equations associated with the generalized 1-parameter Hermite–Frobenius Euler poly- nomials. Mathematical and Computer Modelling of Dynamical Systems, 30(1):683– 700, 2024. [32] H. Garg. Some picture fuzzy aggregation operators and their applications to multi- criteria decision-making. Arabian Journal for Science and Engineering, 42(12):5275– 5290, 2017. [33] S. Dogra and M. Pal. Picture fuzzy matrix and its application. Soft Computing, 24:9413–9428, 2020. [34] I. Silambarasan. Some algebraic structures of picture fuzzy matrices. World Scientific News, 150:78–91, 2020. [35] P. Murugadas. Implication operation on picture fuzzy matrices. AIP Conference Proceedings, 2375:020008, 2021. [36] Tahir Mahmood, Kifayat Ullah, Qaisar Khan, and Naeem Jan. An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy R. A. Padder et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6095 24 of 24 sets. Neural Computing and Applications, 31:7041–7053, 2019. [37] U. Kifayat, T. Mahmood, and N. Jan. Similarity measures for T-spherical fuzzy sets with applications in pattern recognition. Symmetry, 10(6):193, 2018. [38] I. Silambarasan. Spherical fuzzy matrices. TWMS Journal of Applied and Engineering Mathematics, 13(1):98–109, 2023. [39] V. Kumar, Anjana Gupta, and H. C. Taneja. Interval valued picture fuzzy matrix: basic properties and application. Soft Computing, 27:14929–14950, 2023.