Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 193 https://internationalpubls.com The Bounds of Energies of Rough Complemented Graph B.Praba1, B.Sudha1, Aathish sivasubrahmanian2 1,2 Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Chennai - 603110, India. prabab@ssn.edu.in, sudhab@ssn.edu.in, aathish04@gmail.com Article History: Received: 27-05-2024 Revised: 20-07-2024 Accepted: 30-07-2024 Abstract: The main objective of this paper is to study the various energies and their bounds of the Rough complemented graph corresponding to the given Rough semiring. In this paper, for a given approximation space I=(U,R) where U is the nonempty finite set of objects and R is an equivalence relation on U, the Rough semiring (T,∆,∇) is taken for study. The Rough complemented graph of T denoted by GRC(T) is a graph whose vertices are V(GRC(T))={RS(Y)|Y∈〖℘(E)〗^1 } be the set of equivalence classes induced by I and two distinct vertices RS(X) and RS(Y) are adjacent iff RS(X)∇RS(Y)=RS(∅). Note that there will be 2^n-2 vertices in GRC(T). Also Randic, seidel, minimum dominating, maximal independent and dominating energies of GRC(T) are obtained, the lower and upper bounds of these energies are also established. These energies are obtained through Python programming, and a bar diagram is used to conduct a comparative study for various values of n. All the illustrated concepts are explained with suitable examples. Keywords: Independent dominating set, Minimum dominating energy, Randic energy, Siedel energy, Python code. 1. Introduction The concept of energy of a graph was introduced by I. Gutman [1] in the year 1978. In [4] Rajesh kanna et al. compute Milovanovic bounds of the cocktail party graph and crown graph. Different results on independent dominance in graphs are being examined by the authors [2]. A molecular structure descriptor called Randic index was created by Milan Randi in 1975[6]. Later S.B. Bozkurt et al [7] defined Randic matrix and Randic energy. Further discussion on Randic energy can be found in [3], [5]. In [8] the authors find the minimum dominating energy for various graphs like complete graph, star graph etc. This paper is organized as follows. Section 2 is about preliminaries. In section 3, we introduced the Rough complemented graph 𝐺𝑅𝐶(𝑇) and explore the properties of the graph. Also defined various energies like minimal dominating, seidel, randic etc and look at further bounds. In section 4, we provided the python coding for the corresponding graph energies and conclude in section 5. 2. Preliminaries In this section, the basic definitions required to study the article are listed. Definition 2.1. Let 𝐺 = (𝑉(𝐺), 𝐸(𝐺)) be a simple graph. A set 𝐷 ⊆ 𝑉(𝐺) is s said to be a dominating set if every vertex in 𝑉(𝐺) − 𝐷 is adjacent to atleast one vertex in 𝐷. The domination number of 𝐺, denoted by 𝛾(𝐺), is the minimum cardinality among all dominating sets of 𝐺. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 194 https://internationalpubls.com Definition 2.2. A set is independent if no two vertices in it are adjacent. An independent dominating set of 𝐺 is a set that is both dominating and independent in 𝐺 .The independent domination number of 𝐺 denoted by 𝛼, is the minimum size of an independent dominating set. Lemma 2.1. [𝟓] The Randic spectral radius 𝜌1(𝐺) = 1. Lemma 2.2. [𝟑] If 𝐺 posseses isolated vertices, then 𝑑𝑒𝑡𝑅 = 𝑑𝑒𝑡𝐴 = 0. If 𝐺 does not possess isolated vertices, then 𝑑𝑒𝑡𝑅 = 1 𝑑1𝑑2……………𝑑𝑛 𝑑𝑒𝑡𝐴. Let 𝐼 = (𝑈, 𝑅) be an approximation space where 𝑈 is the nonempty finite set of objects and 𝑅 is an equivalence relation on 𝑈 and for any 𝑥 ∈ 𝑈, [𝑥]𝑅 = {𝑦 ∈ 𝑈|(𝑥, 𝑦) ∈ 𝑅} is said to be an equivalence class. For 𝑋 ⊆ 𝑈, 𝑅𝑆(𝑋) = (𝑅−(𝑋), 𝑅−(𝑋)) be the rough set where R−(X) = {x ∈ U|[x]R X} is said to be a lower approximation space and the upper approximation space is defined as R−(X) = {x ∈ U|[x]R ∩ X ≠ ∅}. Also we defined the set of rough sets as T = {RS(X)|X U}. It has been established that if 𝐼 = (𝑈, 𝑅), (𝑇, ∆, ∇) is a Rough semiring [10]. The partition created by 𝑅 on 𝑈 should consist of {𝑋1, 𝑋2, … … 𝑋𝑚, 𝑋𝑚+1 … … . 𝑋𝑛} where |𝑋𝑖| > 1, 1 ≤ 𝑖 ≤ 𝑚, |𝑋𝑗| = 1, 𝑚 + 1 ≤ 𝑗 ≤ 𝑛. Definition 2.3 [𝟏𝟏] Rough Zero Divisor Graph of Rough Semiring The zero divisor graph of the Rough Semiring (T, ∆, ∇) is T(G) = (V, E) where V is the set of vertices in T(G) consists of nonempty zero divisors i.e, V = {RS(X) ∈ T|RS(X) ≠ RS(∅) is a zero divisor of T} and E is the set of edges connecting the elements of V such that there is an edge connecting RS(X) and RS(Y) in V iff RS(X)∇RS(Y) = RS(∅). This graph T(G) is called a Rough zero divisor graph of the Rough Semiring (T, ∆, ∇). 3. Bounds of various energies of Rough complemented graph Throughout this section, we consider an approximation space 𝐼 = (𝑈, 𝑅) along with the Rough semiring (𝑇, ∆, ∇). Let 𝐸 = {𝑋1, 𝑋2, … … … … . 𝑋𝑛} be the equivalence classes induced by 𝑅 in which {𝑋1, 𝑋2, … … … … . 𝑋𝑚} are the equivalence classes with cardinality greater than 1. In this section, the Rough complemented graph 𝐺𝑅𝐶(𝑇) of the Rough Semiring is introduced. Properties of 𝐺𝑅𝐶(𝑇) is studied and the various energies along with their bounds are dealt in detail. 3.1 Minimum dominating energy Definition 3.1. Rough complemented graph Let (𝑇, ∆, ∇) be a Rough semiring. The Rough complemented graph of 𝑇 denoted by 𝐺𝑅𝐶(𝑇) is a graph whose vertices are 𝑉(𝐺𝑅𝐶(𝑇)) = {𝑅𝑆(𝑌)|𝑌 ∈ (℘(𝐸))1} where (℘(𝐸))1 = ℘(𝐸) − {𝑅𝑆(𝑈), 𝑅𝑆(∅)} and two distinct vertices 𝑅𝑆(𝑋), 𝑅𝑆(𝑍) are adjacent iff 𝑅𝑆(𝑋)∇𝑅𝑆(𝑍) = 𝑅𝑆(∅). Remarks. It is to be noted that • The number of vertices in Rough complemented graph 𝐺𝑅𝐶(𝑇) is 2𝑛 − 2 where 𝑛 denotes the number of equivalence classes in 𝐸. • The number of edges in Rough complemented graph 𝐺𝑅𝐶(𝑇) is 1 2 (3𝑛 − 2𝑛+1 + 1). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 195 https://internationalpubls.com • For any 𝑅𝑆(𝑋) ∈ 𝑉(𝐺𝑅𝐶(𝑇)), the degree of 𝑅𝑆(𝑋) is 2𝑛−𝑟 − 1 where 1 ≤ 𝑟 < 𝑚. • For any 𝑅𝑆(𝑋), 𝑅𝑆(𝑌) ∈ 𝑉(𝐺𝑅𝐶(𝑇)), 𝑑2(𝑅𝑆(𝑋)) = {𝑅𝑆(𝑌) ∈ 𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋), 𝑅𝑆(𝑌) = 2)} |𝑑2(𝑅𝑆(𝑋))| = 2𝑟 − 2 + (2𝑟 − 1)(2𝑛−𝑟 − 2). • For any 𝑅𝑆(𝑋), 𝑅𝑆(𝑌) ∈ 𝑉(𝐺𝑅𝐶(𝑇)), 𝑑3(𝑅𝑆(𝑋)) = {𝑅𝑆(𝑌) ∈ 𝑉(𝐺𝑅𝐶(𝑇))|𝑑(𝑅𝑆(𝑋), 𝑅𝑆(𝑌) = 3)} |𝑑3(𝑅𝑆(𝑋))| = 2𝑟 − 2. Here 𝑑 represents the distance. • The diameter of 𝐺𝑅𝐶(𝑇)is 3. Definition 3.2. Minimum dominating set Consider the Rough complemented graph 𝐺𝑅𝐶(𝑇) = (𝑉(𝐶(𝑇)), 𝐸(𝐶(𝑇))). A subset 𝐷(𝐶(𝑇)) is called the dominating set of 𝐺𝑅𝐶(𝑇) if every vertex in 𝑉(𝐶(𝑇)) − 𝐷(𝐶(𝑇)) is adjacent to some vertex in 𝐷(𝐶(𝑇)). The minimum cardinality of a dominating set 𝐷(𝐶(𝑇)) is called the domination number of the graph 𝐺𝑅𝐶(𝑇), denoted by 𝛾(𝐶(𝑇)). Definition 3.3. For every 𝑅𝑆(𝑋), 𝑅𝑆(𝑌) ∈ 𝑉(𝐶(𝑇)), the minimum dominating matrix is 𝐴𝐷(𝐶(𝑇)) = { 1 𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌) = 𝑅𝑆(∅) 1 𝑖𝑓 𝑅𝑆(𝑋) = 𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋) ∈ 𝐷(𝐶(𝑇)) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Definition 3.4. Minimum dominating energy The minimum dominating energy of 𝐺𝑅𝐶(𝑇)is defined as Ԑ(𝐷(𝐶(𝑇))) = ∑ |𝜇𝑖| 2𝑛−2 𝑖=1 where 𝜇1, 𝜇2 … … … . . 𝜇2𝑛 −2 are the spectrum of 𝐴𝐷(𝐶(𝑇)). The spectral radii of 𝐴𝐷(𝐶(𝑇)) are in non- increasing order i.e., 𝜇1  𝜇2 … … … …  𝜇2𝑛 −2. Theorem 3.1. In 𝐺𝑅𝐶(𝑇), the minimum dominating set is 𝐷(𝐶(𝑇)) = {𝑅𝑆(𝑋𝑖)|𝑖 = 1,2, … … … . . 𝑛} Proof. Let 𝑅𝑆(𝑌) ∈ 𝑉(𝐶(𝑇)) − 𝐷(𝐶(𝑇)). Then the edge set 𝜉 = {(𝑅𝑆(𝑌), 𝑅𝑆(𝑍))|𝑍 ∈ 𝐸 − 𝑌} Next to prove that 𝐷(𝐶(𝑇)) is the minimum dominating set. Note that if we remove any 𝑅𝑆(𝑋𝑖), 𝑖 = 1,2, … … … . . 𝑛 from 𝐷(𝐶(𝑇)) then 𝑅𝑆(𝑋1𝑋2 … … … . 𝑋𝑖−1𝑋𝑖+1 … … … . . 𝑋𝑛) will not be adjacent to any of the elements in 𝐷(𝐶(𝑇)) − 𝑅𝑆(𝑋𝑖) which will affect the dominating property. Hence removal of any element from 𝐷(𝐶(𝑇)) will affect the dominating property. Therefore 𝐷(𝐶(𝑇)) = {𝑅𝑆(𝑋𝑖)|𝑖 = 1,2, … … … . . 𝑛} is the minimum dominating set. Theorem 3.2. Let 𝐺𝑅𝐶(𝑇) be the Rough complemented graph and 𝐴𝐷(𝐶(𝑇)) be the minimum dominating matrix of 𝐺𝑅𝐶(𝑇) and 𝛾(𝐶(𝑇)) denotes the minimum domination number then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 196 https://internationalpubls.com • ∑ 𝜇𝑖 = |𝛾(𝐶(𝑇))| 2𝑛−2 𝑖=1 • ∑ 𝜇𝑖 2𝑛−2 𝑖=1 2 = |3𝑛 − 2𝑛+1 + 1| + |𝛾(𝐶(𝑇))| Proof. It is known that the sum of eigen values of 𝐴𝐷(𝐶(𝑇)) is the trace of 𝐴𝐷(𝐶(𝑇)). ∑ 𝜇𝑖 2𝑛−2 𝑖=1 = ∑ 𝑑𝑖𝑖 2𝑛−2 𝑖=1 = |𝛾(𝐶(𝑇))| It is find that ∑ 𝜇𝑖 22𝑛−2 𝑖=1 of 𝐴𝐷(𝐶(𝑇)) is the trace of [𝐴𝐷(𝐶(𝑇))]2 ∑ 𝜇𝑖 2 2𝑛−2 𝑖=1 = ∑ 𝑑𝑖𝑗 2𝑛−2 𝑖=1 ∑ 𝑑𝑗𝑖 2𝑛−2 𝑗=1 = ∑ 𝑑𝑖𝑖 2 +2𝑛−2 𝑖=1 ∑ 𝑑𝑖𝑗𝑑𝑗𝑖𝑖≠𝑗 = ∑ 𝑑𝑖𝑖 2 + 2 ∑ 𝑑𝑖𝑗 2 𝑖<𝑗 2𝑛−2 𝑖=1 = |𝛾(𝐶(𝑇))| + |(3𝑛 − 2𝑛+1 + 1)| Theorem 3.3. In 𝐺𝑅𝐶(𝑇), 𝐷(𝐶(𝑇)) be the minimum dominant set and ∆𝐷 denote the determinant of 𝐴𝐷(𝐶(𝑇)) then √(3𝑛 − 2𝑛+1 + 1 + 𝛾(𝐶(𝑇))) + (2𝑛 − 2)(2𝑛 − 3)∆𝐷 2 (2𝑛−2)(2𝑛−3)  Ԑ𝐷(𝐺𝑅𝐶(𝑇) )  √(2𝑛 − 2) (3𝑛 − 2𝑛+1 + 1 + 𝛾(𝐶(𝑇))) Proof. By Cauchy Schwarz inequality ( ∑ |𝜇𝑖 2𝑛−2 𝑖=1 |) 2 ≤ ( ∑ 1 2𝑛−2 𝑖=1 ) ( ∑ 𝜇𝑖 2 2𝑛−2 𝑖=1 ) [Ԑ𝐷(𝐺𝑅𝐶(𝑇 )) ] 2 ≤ (2𝑛 − 2) (3𝑛 − 2𝑛+1 + 1 + 𝛾(𝐶(𝑇))) And from arithmetic –geometric mean inequality ∑ |𝜇𝑖||𝜇𝑗| ≥ (2𝑛 − 2)(2𝑛 − 3)∆𝐷 2 (2𝑛−2)(2𝑛−3) 𝑖≠𝑗 [Ԑ𝐷(𝐺𝑅𝐶(𝑇 )) ] 2 = (∑ |𝜇𝑖| 2𝑛−2 𝑖=1 ) 2 = ∑ |𝜇𝑖| 2𝑛−2 𝑖=1 2 + ∑ |𝜇𝑖||𝜇𝑗𝑖≠𝑗 | [Ԑ𝐷(𝐺𝑅𝐶(𝑇 )) ] 2 ≥ (3𝑛 − 2𝑛+1 + 1 + 𝛾(𝐶(𝑇))) + (2𝑛 − 2)(2𝑛 − 3)∆𝐷 2 (2𝑛−2)(2𝑛−3) Ԑ𝐷(𝐺𝑅𝐶(𝑇)) ≥ √(3𝑛 − 2𝑛+1 + 1 + 𝛾(𝐶(𝑇))) + (2𝑛 − 2)(2𝑛 − 3)∆𝐷 2 (2𝑛−2)(2𝑛−3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 197 https://internationalpubls.com 3.2. Maximal independent and dominating energy Definition 3.5. The maximal independent and dominating set of the Rough complemented graph 𝐺𝑅𝐶(𝑇) is denoted by 𝐼𝐷(𝐶(𝑇)). Note that the elements of 𝐼𝐷(𝐶(𝑇)) will be a maximum independent set as well as a dominating set. Definition 3.6. Let 𝐼𝐷(𝐶(𝑇)) be a maximal independent and dominating set of a graph 𝐺𝑅𝐶(𝑇), then the corresponding adjacency matrix is denoted by 𝐴𝐼𝐷(𝐶(𝑇)) is defined as 𝐴𝐼𝐷(𝐶(𝑇)) = { 1 𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌) = 𝑅𝑆(∅) 1 𝑖𝑓 𝑅𝑆(𝑋) = 𝑅𝑆(𝑌)𝑎𝑛𝑑𝑅𝑆(𝑋) ∈ 𝐼𝐷(𝐶(𝑇)) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Definition 3.7. The maximal independent and dominating energy of 𝐺𝑅𝐶(𝑇) is defined by Ԑ𝐼𝐷(𝐶(𝑇)) = ∑ |𝜔𝑖| 2𝑛−2 𝑖=1 where 𝜔𝑖 are the eigen values of 𝐴𝐼𝐷(𝐶(𝑇)) and are in non - increasing order. Theorem 3.4. For the Rough complemented graph 𝐺𝑅𝐶(𝑇), |𝐼𝐷(𝐶(𝑇))| is 2𝑛−1 − 1. Proof. Consider 𝐼𝐷(𝐶(𝑇)) = {𝑅𝑆(𝑋𝑖 ∪ 𝑌)|𝑌 ∈ ℘(𝐸 − 𝑋𝑖)} To prove 𝐼𝐷(𝐶(𝑇)) is an independent set. Let 𝑅𝑆(𝑋), 𝑅𝑆(𝑍) ∈ 𝐼𝐷(𝐶(𝑇)), where 𝑅𝑆(𝑋) = 𝑅𝑆(𝑋𝑖 ∪ 𝑌1) and𝑅𝑆(𝑍) = 𝑅𝑆(𝑋𝑖 ∪ 𝑌2) where 𝑌1, 𝑌2 ∈ ℘(𝐸 − 𝑋𝑖) This implies 𝑅𝑆(𝑋𝑖) ∈ 𝑅𝑆(𝑋)∇𝑅𝑆(𝑍) ≠ 𝑅𝑆(∅) There is no edge between 𝑅𝑆(𝑋)𝑎𝑛𝑑 𝑅𝑆(𝑍) ∴ 𝐼𝐷(𝐶(𝑇)) is an independent set. It is clear to verify that addition of any vertex to 𝐼𝐷(𝐶(𝑇)) will affect the independence property. Hence 𝐼𝐷(𝐶(𝑇)) is the maximal independent set. Next to prove that 𝐼𝐷(𝐶(𝑇)) is a dominating set. Let 𝑅𝑆(𝑍) ∈ 𝑉(𝐺𝑅𝐶(𝑇)) − 𝐼𝐷(𝐶(𝑇)) Since the elements of 𝑉(𝐺𝑅𝐶(𝑇)) − 𝐼𝐷(𝐶(𝑇)) are from ℘(𝐸 − 𝑋𝑖) and so 𝑅𝑆(𝑍) is adjacent to 𝑅𝑆(𝑋𝑖) and hence it is a dominating set. Hence it is clear that |𝐼𝐷(𝐶(𝑇))| = 2𝑛−1 − 1. Remarks 3.1. For the Rough complemented graph 𝐺𝑅𝐶(𝑇), |𝐼𝐷(𝐶(𝑇))| ≤ 2𝑛 − 3. Also |𝐼𝐷(𝐶(𝑇))| ≥ 2𝑛−2 𝑛 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 198 https://internationalpubls.com 3.3. Siedel Energy Definition 3.8. The Siedel matrix of 𝐺𝑅𝐶(𝑇) is defined by 𝑆(𝐶(𝑇)) = { −1 𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌) = 𝑅𝑆(∅) 1 𝑅𝑆(𝑋)∇𝑅𝑆(𝑌) ≠ 𝑅𝑆(∅) 0 𝑖𝑓 𝑅𝑆(𝑋) = 𝑅𝑆(𝑌) The Siedel energy of 𝐺𝑅𝐶(𝑇) is defined as 𝑆𝐸(𝐶(𝑇)) = ∑ | i |2n−2 i=1 , 𝑖 = 1,2 … … … 2n − 2, where i s are the eigen values of 𝑆(𝐶(𝑇)). Lemma 3.1. Let 𝜆1, 𝜆2 … … … … . . 𝜆2𝑛−2 denote the siedel eigenvalues of 𝑆(𝐶(𝑇)) then • ∑ 𝑖 2𝑛−2 𝑖=1 = 0 • ∑ 𝑖 2 = (2𝑛 − 2)( 2𝑛 − 3)2𝑛−2 𝑖=1 Proof. It is known that ∑ 𝑖 2𝑛−2 𝑖=1 = ∑ 𝑠𝑖𝑖 2𝑛−2 𝑖=1 = 0 Sum of squares of eigen values of 𝑆(𝐶(𝑇)) is the trace of (𝑆(𝐶(𝑇))) 2 . ∑ 𝑖 2 = 2𝑛−2 𝑖=1 ∑ 𝑠𝑖𝑗 2𝑛−2 𝑖=1 ∑ 𝑠𝑗𝑖 2𝑛−2 𝑗=1 = ∑ (𝑠𝑖𝑖) 2 2𝑛−2 𝑖=1 + ∑ 𝑠𝑖𝑗𝑠𝑗𝑖 𝑖≠𝑗 = ∑ (𝑠𝑖𝑖) 22𝑛−2 𝑖=1 + 2 ∑ 𝑠𝑖𝑗 2 𝑖<𝑗 = 2 { 1 2 (3𝑛 − 2𝑛+1 + 1)(−1)2 + ( (2𝑛−2)2 –(2𝑛−2) 2 − 1 2 (3𝑛 − 2𝑛+1 + 1)) (1)2} = (2𝑛 − 2)2 − (2𝑛 − 2) ∑ 𝑖 2 2𝑛−2 𝑖=1 = 2𝑛(2𝑛 − 5) + 6 Theorem 3.5. In 𝐺𝑅𝐶(𝑇), ∆𝑆= |𝑑𝑒𝑡𝑆(𝐶(𝑇))| then √2𝑛(2𝑛 − 5) + 6 + (2𝑛 − 2)(2𝑛 − 3)∆S 2 2𝑛−2≤ 𝑆𝐸(𝐶(𝑇)) ≤ √(2𝑛 − 2){2𝑛(2𝑛 − 5) + 6} where |𝑑𝑒𝑡𝑆(𝐶(𝑇))| means the absolute value. Proof. Taking 𝑎𝑖 = 1, 𝑏𝑖 = |𝑖| Cauchy Schwarz inequality becomes (∑ |𝑖| 2𝑛−2 𝑖=1 ) 2 ≤ (∑ 12𝑛−2 𝑖=1 )(∑ 𝑖 22𝑛−2 𝑖=1 ) (𝑆𝐸(𝐶(𝑇))) 2 ≤ (2𝑛 − 2)(2𝑛(2𝑛 − 5) + 6) (𝑏𝑦 𝑙𝑒𝑚𝑚𝑎 3.1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 199 https://internationalpubls.com 𝑆𝐸(𝐶(𝑇)) ≤ √(2n − 2)(2n(2n − 5) + 6) By arithmetic geometric mean inequality ∑ | i || j | ≥ (2𝑛 − 2)(2𝑛 − 3)∆S 2 2𝑛−2i≠j (𝑆𝐸(𝐶(𝑇))) 2 = (∑ |i| 2n−2 i=1 ) 2 = ∑ |i| 2n−2 i=1 2 + ∑ | i || j |i≠j (𝑆𝐸(𝐶(𝑇))) 2 ≥ 2𝑛(2𝑛 − 5) + 6 + (2𝑛 − 2)(2𝑛 − 3)∆S 2 2𝑛−2 𝑆𝐸(𝐶(𝑇)) ≥ √(2𝑛 − 2)(2𝑛 − 3) (1 + ∆S 2 2𝑛−2) 3.4. Randic Energy Definition 3.9. The Randic matrix 𝑅(𝐶(𝑇)) = 𝑅𝑥𝑦 of 𝐺𝑅𝐶(𝑇) is a square matrix of order 2𝑛 − 2 whose (𝑥, 𝑦) entry is 𝑅𝑥𝑦 = { 1 √𝑑𝑥𝑑𝑦 𝑖𝑓𝑅𝑆(𝑋)∇𝑅𝑆(𝑌) = 𝑅𝑆(∅) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Where 𝑑𝑥 denote the degree of the vertex 𝑅𝑆(𝑋). The eigenvalues of 𝑅(𝐶(𝑇)) are called Randic eigenvalues and are denoted by 𝜌1, 𝜌2 … … … . . 𝜌2𝑛−2. If all the 𝜌𝑖’s , 1 ≤ 𝑖 ≤ 2𝑛 − 2 are distinct then the Randic spectrum of 𝐺𝑅𝐶(𝑇) can be denoted as 𝑆𝑝𝑒𝑐 (𝑅(𝐶(𝑇))) = ( 𝜌1 𝜌2 … … … . . 𝜌2𝑛−2 𝑚1 𝑚2 … … … 𝑚2𝑛−2 ) Where 𝑚𝑗 indicates the algebraic multiplicity of the eigenvalue  𝑗 , 1 ≤ 𝑗 ≤ 2𝑛 − 2 of 𝐺𝑅𝐶(𝑇). The Randic energy of 𝐺𝑅𝐶(𝑇)is defined as 𝑅𝐸(𝐶(𝑇)) = ∑ | 𝑖 |2𝑛−2 𝑖=1 , 𝑖 = 1,2, … … … . 2𝑛 − 2. Theorem 3.6. For 𝐺𝑅𝐶(𝑇), 𝑅𝐸(𝐶(𝑇)) ≤ 1 + √ (2𝑛−3)(2𝑛−2−(𝐶(𝑇)) (𝐶(𝑇)) where (𝐶(𝑇)) is the minimum degree. Proof. It is known that ∑ 𝜌𝑖 22𝑛−2 𝑖=1 = 2 ∑ 1 𝑑𝑥𝑑𝑦 𝑥,𝑦∈𝐸(𝐶(𝑇)) = ∑ 1 𝑑𝑥 2𝑛−2 𝑥=1 ∑ 1 𝑑𝑦 𝑥,𝑦∈𝐸(𝐶(𝑇)) ≤ ∑ 1 (𝐶(𝑇)) 2𝑛−2 𝑥=1 ∑ 1 𝑑𝑦 𝑥,𝑦∈𝐸(𝐶(𝑇)) 𝑆𝑖𝑛𝑐𝑒 𝑑𝑥 ≥ (𝐶(𝑇)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 200 https://internationalpubls.com = 2𝑛 − 2 𝑅𝐸(𝐶(𝑇)) = ∑ | 𝑖 | = 1 + ∑ | 𝑖 |2𝑛−2 𝑖=2 2𝑛−2 𝑖=1 (𝑏𝑦 𝑙𝑒𝑚𝑚𝑎 2.1) ≤ 1 + √(2𝑛 − 3)(∑ 𝜌𝑖 2 − 12𝑛−2 𝑖=1 ) (by Cauchy Schwarz inequality) ≤ 2𝑛 − 2 Theorem 3.7. In 𝐺𝑅𝐶(𝑇), with maximum degree ( C(T)) 𝑅𝐸(𝐶(𝑇)) ≥ 1 + √ 2𝑛 − 2 ( 𝐶(𝑇)) − 1 + (2𝑛 − 3)(2𝑛 − 4) ( | det 𝐴(𝐶(𝑇))| ∏ 𝑑𝑖 2𝑛−2 𝑖=1 ) 2 2𝑛−3 Where 𝑑𝑒𝑡 𝐴(𝐶(𝑇)) denotes the determinant of adjacency matrix of 𝐺𝑅𝐶(𝑇). Proof. Proceeding as in the above, we have ∑ 𝜌𝑖 2𝑛−2 𝑖=1 2 = 2𝑛−2 (𝐶(𝑇)) Using arithmetic geometric mean inequality 2 ∑ |𝜌𝑖||𝜌𝑗| ≥ 2≤𝑖<𝑗≤2𝑛−2 (2𝑛 − 3)(2𝑛 − 4) ( ∏ |𝜌𝑖| 2𝑛−2 𝑖=2 ) 2 2𝑛−3 = (2𝑛 − 3)(2𝑛 − 4)(|det RE(𝐶(𝑇)) |) 2 2𝑛−3 = (2𝑛 − 3)(2𝑛 − 4) ( | det 𝐴(𝐶(𝑇))| ∏ 𝑑𝑖 2𝑛−2 𝑖=1 ) 2 2𝑛−3 (𝑏𝑦 𝑙𝑒𝑚𝑚𝑎 2.2) Now, (∑ |𝜌𝑖 2𝑛−2 𝑖=2 |) 2 = ∑ 𝜌𝑖 2𝑛−2 𝑖=2 2 + 2 ∑ |𝜌𝑖||𝜌𝑗|2≤𝑖<𝑗≤2𝑛−2 ∑ | 𝑖 | 2𝑛−2 𝑖=2 ≥ √ 2𝑛 − 2 (𝐶(𝑇)) − 1 + (2𝑛 − 3)(2𝑛 − 4) ( | det 𝐴(𝐶(𝑇))| ∏ 𝑑𝑖 2𝑛−2 𝑖=1 ) 2 2𝑛−3 𝑅𝐸(𝐶(𝑇)) = ∑ | 𝑖 |2𝑛−2 𝑖=1 𝑅𝐸(𝐶(𝑇)) ≥ 1 + √ 2𝑛−2 (𝐶(𝑇)) − 1 + (2𝑛 − 3)(2𝑛 − 4) ( | det 𝐴(𝐶(𝑇))| ∏ 𝑑𝑖 2𝑛−2 𝑖=1 ) 2 2𝑛−3 Example 3.1. The following graph is corresponding to the approximation space 𝐼 = (𝑈, 𝑅) where 𝑅 induces 3 equivalence classes. Let 𝑈 = (𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6) Here 𝐸 = {𝑋1, 𝑋2, 𝑋3} 𝑤ℎ𝑒𝑟𝑒 𝑋1 = {𝑥1, 𝑥3}, 𝑋2 = {𝑥2, 𝑥4, 𝑥6} , 𝑋3 = {𝑥5} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 201 https://internationalpubls.com 𝐻𝑒𝑟𝑒 𝑉(𝐶(𝑇)) = { 𝑅𝑆(𝑋1), 𝑅𝑆(𝑋2), 𝑅𝑆(𝑋3), 𝑅𝑆(𝑥1), 𝑅𝑆(𝑥2), 𝑅𝑆(𝑥1 ∪ 𝑋2), 𝑅𝑆(𝑋1 ∪ 𝑥2), 𝑅𝑆(𝑥1 ∪ 𝑥2), 𝑅𝑆(𝑋1 ∪ 𝑋2), 𝑅𝑆(𝑥1 ∪ 𝑋3), 𝑅𝑆(𝑋1 ∪ 𝑋3), 𝑅𝑆(𝑥2 ∪ 𝑋3), 𝑅𝑆(𝑋2 ∪ 𝑋3), 𝑅𝑆(𝑥1 ∪ 𝑋2 ∪ 𝑋3), 𝑅𝑆(𝑋1 ∪ 𝑥2 ∪ 𝑋3), 𝑅𝑆(𝑥1 ∪ 𝑥2 ∪ 𝑋3)} The minimum dominating matrix 𝐴𝐷(𝐶(𝑇)) is given by 𝐴𝐷(𝐶(𝑇)) = Figure 1: Rough complemented graph The spectrum of minimum dominating energy are [3.3028, 1(2), −0.3028, (−1)(2)] The maximal independent and dominating matrix of 𝐺𝑅𝐶(𝑇) is given by The characteristic polynomial, spectrum and maximal independent domination energy are as follows. f(𝐺𝑅𝐶(𝑇),) = (2 − 2)(3 − 32 −  + 4) Spec (𝐺𝑅𝐶(𝑇)) = (−1.4142 1 −1.1149 1 0 1 1.2541 1 1.4142 1 2.8608 1 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 202 https://internationalpubls.com Ԑ𝐼𝐷(𝐶(𝑇)) = 8.0582. Here the maximal independent and domination number |𝐼𝐷(𝐶(𝑇))| = 2n−1 − 1 = 3. 4. Generation of various graph energies using Python for the Rough complemented graph In this section, Python Code is provided for Siedel, Randic and Minimum dominating energies for the graph 𝐺𝑅𝐶(𝑇). Additionally, the energies are compared using bar diagram for various values of 𝑛. Certain auxiliary functions used for displaying and increasing the visual appeal of the graphs and their energies are not included in the code presented. Libraries used to aid code reusability have been imported at the top of the first code section. Graph generation code import numpy as np import networkx as nx from itertools import chain, combinations import matplotlib.pyplot as plt import csv # Utility Functions def powerset(iterable): s = set(iterable) return set(chain.from_iterable(combinations(s, r) for r in range(len(s)+1))) def EnergyOfMatrix(A): eigVals = np.linalg.eigvals(A) # Compute eigenvalues of A sumAbsEigVals = sum(abs(eigVals)) # Compute sum of absolute values of eigenvalues return sumAbsEigVals def EnergyOfGraph(G): return EnergyOfMatrix(nx.adjacency_matrix(G).todense()) n = int(input("Enter the value of n: ")) n_nat = set(range(1, n + 1)) powerset_n_nat = powerset(n_nat) powerset_n_nat_min_1 = powerset(range(2, n + 1)) i_d_set = set( # Independent Dominating Set (1, x) for x in powerset_n_nat_min_1.difference(set(range(2, n + 1))) ) vertices = set( elem for elem in powerset(n_nat) if (elem not in [tuple(), tuple(n_nat)])) edges = set( Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 203 https://internationalpubls.com (vert1, vert2) for vert1 in vertices for vert2 in vertices if ( (len(set(vert1).intersection(set(vert2))) == 0) or (vert1 == vert2 and (vert1 in i_d_set)) ) ) # %% graph = nx.Graph() graph.add_nodes_from(vertices) graph.add_edges_from(edges) graphdegrees = {node: val for (node, val) in graph.degree()} siedel_edges = [] # Stores edges as randic_edges = [] # (v1,v2,weight) triples for vert1 in vertices: for vert2 in vertices: if set(vert1) == set(vert2): if len(vert1) == 1: pass elif len(set(vert1).intersection(set(vert2))) == 0: siedel_edges.append((vert1, vert2, -1)) # min_dom.append((vert1,vert2,1)) randic_edges.append( (vert1, vert2, 1 / np.sqrt((graphdegrees[vert1] * graphdegrees[vert2]))) ) elif len(set(vert1).intersection(set(vert2))) != 0:# min_dom.append((vert1,vert2,0)) siedel_edges.append((vert1, vert2, 1)) randic_edges.append((vert1, vert2, 0)) seidel_graph = nx.Graph() seidel_graph.add_weighted_edges_from(siedel_edges) randic_graph = nx.Graph() randic_graph.add_weighted_edges_from(randic_edges) min_dom_adj = nx.adjacency_matrix(graph).todense() # GRC T for i in range(len(min_dom_adj)): for j in range(len(min_dom_adj[0])): if i==j: if len(list(vertices)[i]) == 1: min_dom_adj[i][i] = 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 204 https://internationalpubls.com min_dom_graph = nx.relabel_nodes(nx.from_numpy_array(min_dom_adj),{i:vert for i,vert in enumerate(vertices)}) # mapping = {i:vert for i,vert in enumerate(vertices)} Energy Calculation Code Functions were written that would ease the process of calculating the Graph energies for any graph: def EnergyOfMatrix(A): eigVals = np.linalg.eigvals(A) # Compute eigenvalues of A sumAbsEigVals = sum(abs(eigVals)) # Compute sum of absolute values of eigenvalues return sumAbsEigVals def EnergyOfGraph(G): return EnergyOfMatrix(nx.adjacency_matrix(G).todense()) Generated Graphs and Charts Fig 2: 𝐺𝑅𝐶(𝑇) graph when 𝑛 = 4 Fig 3: The Siedel Graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 205 https://internationalpubls.com Fig 4: The Randic Graph Fig 5: Minimum dominating graph Fig 6: A comparison of the Adjacency Energy, Siedel Energy, Randic Energy and Minimum Dominating Energy for 𝐺𝑅𝐶(𝑇) graph for varying values of 𝑛. 5. Conclusion In this study, the Rough complemented graph of the Rough semiring is defined using the equivalence classes. Also the maximal independent and dominating set of 𝐺𝑅𝐶(𝑇) graph is established. Additionally, the minimum dominating, Siedel and Randic energies of the 𝐺𝑅𝐶(𝑇) graph are defined and the lower and upper bounds are derived. All the above mentioned energies can be found using Python and several values of 𝑛 are compared. Each notion is illustrated and supported by an example. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 206 https://internationalpubls.com Conflicts of interest: The authors declares that there is no conflict of interest regarding the publication of this article. Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors Acknowledgement: The authors would express their sincere gratitude to The Management, and The Principal, of Sri Sivasubramaniya Nadar College of Engineering for their constant support. References: [1] I. Gutman, The Energy of a Graph, Ber. Math. Stat. Sekt. Forschungszentrum Graz 103, (1978) 1-22. [2] Wayne Goddard, Michael A. Henning, Independent domination in graphs: A survey and recent results, Discrete Mathematics 313 (2013) 839–854. [3] Ivan Gutman, Boris Furtula ̧S.BurcuBozkurt, On Randic energy, Linear Algebra and its Applications 442 (2014), 50-57. [4] M. R. Rajesh Kanna, R. Pradeep Kumar, Mohammad Reza Farahani, Milovanovic Bounds for Seidel Energy of a Graph, Advances in Theoretical and Applied Mathematics, Volume 10, Number 1 (2016), 37–44. [5] Bolian Liu, Yufei Huang, Jingfang Feng, A Note on the Randic Spectral Radius, MATCH Commun. Math. Comput. Chem. 68 (2012) 913. [6] M. Randic, On Characterization of Molecular Branching, Journal of the American Chemical Society, 97(1975), 6609-6615. [7] S. B. Bozkurt, A. D. Gungor, and I.Gutman, Randic Spectral Radius and Randic Energy, Communications in Mathematical and in Computer Chemistry, 64(2010), 239-250. [8] M. R. Rajesh Kanna, B. N. Dharmendra, G. Sridhara, The Minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics, Volume 85 No. 4 (2013), 707-718. [9] Madhukar M. Pawar, Shilpa T. Bhangale, Minimum independent dominating energy of graphs, Asian-European Journal of Mathematics, (2021). [10] B. Praba, V. M. Chandrasekaran, A. Manimaran, Semiring on rough sets, Ind. J. of Sci and Tech. 3 (2015) 280-286. [11] B. Praba, A. Manimaran, V. M. Chandrasekaran, The zero divisor graph of a Rough semiring, Int. J. Pure and Appl. Math. 98 (2015) 33-37. [12] M.R. Rajesh Kanna, R. Jagadeesh, B.K. Kempegowda, Minimum dominating Seidel energy of a graph, International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016.