EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5971 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bounds on the Energy of Zero-divisor Graph of Quotient Ring and Its Topological Indices Vira Hari Krisnawati1,, Noor Hidayat1, Ayunda Faizatul Musyarrofah∗1 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, East Java, Indonesia Abstract. In this paper, we study the zero-divisor graph of Z℘[x]/⟨x5⟩ for prime number ℘, denoted as Γ(Z℘[x]/⟨x5⟩), including its energy and topological indices. Specifically, we provide bounds of the energy for Γ(Z℘[x]/⟨x5⟩) and show that these bounds are numerically close to the actual energy value. Furthermore, we determine the topological indices of Γ(Z℘[x]/⟨x5⟩), including the topological indices based on distance and degree. We also perform numerical simulations of the topological indices for various prime numbers ℘. 2020 Mathematics Subject Classifications: 13A70, 05C50, 05C09 Key Words and Phrases: zero-divisor graph, graph energy, distance-based topological indices, degree-based topological indices 1. Introduction The concept of graphs regarding different algebraic structures is interesting to investi- gate because it enables us to explore algebraic properties using graph theory. One notable example is the zero-divisor graph that came from the work done by Beck in 1988 [1]. He defined the vertex set as zero divisors, including zero. In 1999, Anderson and Livingston revised the definition by focusing on only non-zero zero divisors as its vertices, denoted as Γ(R) for commutative ring R [2]. Since then, the zero-divisor graph has been a fast- developing area and widely applied in various fields, including algebraic cryptography [3, 4] and coding theory [5, 6]. For further literature on this topic, see [7–12]. Besides its importance in algebra, graph theory has also seen rapid development in its applications to chemistry, particularly through the study of graph energy. In 1978, Gutman first defined the graph energy as the total of the absolute values of its adja- cency matrix’s eigenvalues [13]. This concept arose when Erich Huckel developed Huckel molecular orbital theory to estimate the π-electron energy [14]. Beyond graph energy, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5971 Email addresses: virahari@ub.ac.id (V. H. Krisnawati), noorh@ub.ac.id (N. Hidayat), ayundafaiza02@gmail.com (A. F. Musyarrofah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 2 of 19 topological indices are also important parameters for analyzing molecular structures and predicting chemical properties. Most of the studies of topological indices currently are based on distance or degree. The Wiener index, proposed by Wiener [15] in 1947, is the earliest distance-based topological indices and is used to approximate the boiling points of alkanes. In 1993, Randic [16] proposed the hyper-Wiener index applied for analyz- ing the physicochemical properties of organic compounds. Meanwhile, first degree-based topological indices were proposed in 1970s by Gutman and Trinajstic [17]. There are the first and second Zagreb indices that are used for analyzing the thermodynamic stability and reactivity of unsaturated molecules. Furthermore, in 1984, Narumi and Katayama proposed the Narumi-Katayama index, a simpler degree-based topological index used to examine the branching structures of saturated hydrocarbons[18]. In addition some researchers have extended their study to various algebraic graph structures, including zero-divisor graph. In 2011, Ahmadi and Jahani-Nezhad pioneered the examination of the energy and Wiener index of Γ(Zpq) and Γ(Zp2q) for every distinct prime p, q [19]. Later, Johnson and Sankar in 2023 studied the energy and topological indices of Γ(Zp[x]/⟨x4⟩) for prime number p [20]. However, Rather [21] revised their results on the energy and second Zagreb index formula. Rayer and Jeyaraj in 2023 studied the topological indices of Γ(Zp2 [x]/⟨x2⟩) for every prime q ≥ 3 and Γ(Zpq[x]/⟨x2⟩) for every prime 2 < p < q [22], and further investigated the energy of Γ(Zp2 [x]/⟨x2⟩) in 2024 [23]. Previous research has primarily focused on determining the energy and topological indices of the zero-divisor graph in quotient rings with principal ideals ⟨x2⟩ and ⟨x4⟩. Recently, Musyarrofah et al. [24] explored a different type of quotient ring structure, specifically Γ(Z℘[x]/⟨x5⟩) for a prime number ℘, focusing on the fundamental properties of the graph. However, their study did not examine its energy or topological indices in detail. In this paper, we address this gap by analyzing the energy and topological indices of Γ(Z℘[x]/⟨x5⟩) for prime number ℘. Specifically, we obtain the lower and upper bounds of the energy for the graph and these bounds are numerically close to the actual energy value. This provides a reliable method for estimating the energy of the graph in similar cases. Furthermore, we investigate topological indices, including distance-based topological in- dices such as the Wiener and hyper-Wiener indices, as well as degree-based topological indices such as the first, second Zagreb, and Narumi-Katayama indices. To validate our theoretical results, we conduct numerical simulations using a computer software MAT- LAB, comparing the computed graph energy with its theoretical bounds and analyzing the growth patterns of various topological indices. 2. Preliminaries In this section presents basic concepts, notations, and preliminaries relevant to this paper. All of the graphs mentioned are simple graphs, meaning that they are undirected, do not have loops, and do not contain multiple edges. The fundamental concepts of graph theory discussed in this article are referenced in [25]. Let G be a graph with vertex set V (G) = {v1, v2, v3, . . . , vn}. The adjacency matrix of G, A(G) = [aij ] is a (0, 1)-symmetric V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 3 of 19 matrix with entries aij equal to 1 if vi is adjacent to vj and equal to 0 otherwise. The degree of a vertex v ∈ V (G), deg(v), indicates the number of vertices that are adjacent to v and the distance d(v, w) indicates the shortest path between the vertices v and w. The distance matrix of G, D(G) = [dij ] is a symmetric matrix, where dij denotes the distance between two distinct vertices vi and vj . Zero-divisor graph is the subject in this investigation, where the definition used follows Anderson and Livingston [2]. Suppose R is a commutative ring, with Z(R) being the set of zero-divisors of R. The zero-divisor graph of R, Γ(R) is the graph with the set of vertex consisting of the elements of Z∗(R) = Z(R) \ {0} and the set of edges E(Γ(R)) = {xy | xy = 0, ∀x, y ∈ Z∗(R)}. Furthermore, we present several results related to block and circulant matrices that are utilized. These results are used to calculate the eigenvalues and determinants of the adjacency matrix. Lemma 1. [26] Let P,Q,R, S be matrices and suppose that matrix P is invertible. If M = [ P Q R S ] , then det(M) = det(P ) · det(S −RP−1Q). A circulant matrix is a square matrix where each row is generated by moving the entries of the previous row one position to the right, while the last entry moves to the first position. This matrix C can be represented by the vector c = [c0, c1, c2, . . . , cm−1], where each row results from a right circular shift of c. The circulant matrix of order m×m with entries c0, c1 ∈ R is denoted as C(c0,c1,m) and has the following form: C(c0,c1,m) =  c0 c1 c1 . . . c1 c1 c0 c1 . . . c1 c1 c1 c0 . . . c1 ... ... ... . . . ... c1 c1 c1 . . . c0  m×m . Proposition 1. [27] Let C(c0,c1,m) be a circulant matrix. The determinant of C(c0,c1,m) is det(C(c0,c1,m)) = [c0 + (m− 1)c1](c0 − c1) m−1. Proposition 2. [27] Let C(c0,c1,m) be a nonsingular circulant matrix. The inverse of C(c0,c1,m) is C−1 (c0,c1,m) = 1 det(C(c0, c1,m))  φm−1 ϑm−1 ϑm−1 . . . ϑm−1 ϑm−1 φm−1 ϑm−1 . . . ϑm−1 ϑm−1 ϑm−1 φm−1 . . . ϑm−1 ... ... ... . . . ... ϑm−1 ϑm−1 ϑm−1 . . . φm−1  C−1 (c0,c1,m) = 1 det(C(c0, c1,m)) C(φm−1, ϑm−1,m), V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 4 of 19 where φm−1 = [c0 + (m− 2)c1](c0 − c1) m−2 and ϑm−1 = −c1(c0 − c1) m−2. The study of graph energy proposed by Gutman in 1978 [13], is defined as: En(G) = n∑ i=1 |λi|, where λi represents the eigenvalues of A(G). To further analyze graph energy, the Maclaurin symmetric mean inequality is used to establish lower and upper bounds. Proposition 3. [28] Let a1, a2, a3, . . . , as be a positive real numbers and the average of the product of all subsets of order k represented by ∏ k, where the number of such subsets is given by the number of ways to choose k elements from s elements. The values of ∏ k are defined as follows.∏ 1 = a1 + a2 + a3 + · · ·+ as s ,∏ 2 = 1 s(s−1) 2 (a1a2 + a1a3 + · · ·+ a1as + a2a3 + · · ·+ as−1as), ...∏ s = a1a2a3 . . . as. The Maclaurin symmetric mean inequality states that∏ 1 ≥ ∏ 2 1/2 ≥ ∏ 3 1/3 ≥ ∏ 4 1/4 ≥ · · · ≥ ∏ s 1/s, (1) with equalities holding if and only if a1 = a2 = a3 = · · · = as We review several topological indices of graph that are used in this paper. The Wiener index [15] can be described as the following equation. W(G) = 1 2 n∑ i=1 n∑ j=1 dij . Meanwhile, the Hyper-Wiener index [16] is defined as WW(G) = 1 2 W(G) + 1 4 n∑ i=1 n∑ j=1 (dij) 2. V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 5 of 19 The first Zagreb index and the second Zagreb index [17] are respectively defined as M1(G) = ∑ vw∈E(G) [deg(v) + deg(w)] and M2(G) = ∑ vw∈E(G) [deg(v) deg(w)]. The Narumi-Katayama index [18] is defined as NK(G) = ∏ v∈V (G) deg(v). 3. Energy of zero-divisor graph of Z℘[x]/⟨x5⟩ In this section, we discuss the energy of Γ(Z℘[x]/⟨x5⟩). To provide a comprehensive understanding, we first revisit essential results related to the structure of Γ(Z℘[x]/⟨x5⟩). Next, we explore the calculation of eigenvalues. Following this, we discuss the lower and upper bounds for graph’s energy. Let Z℘[x] be a polynomial commutative ring and ⟨x5⟩ be a principal ideal of Z℘[x]. A quotient ring Z℘[x] is formed from the set of all cosets Z℘[x]/⟨x5⟩ = {kx4 + lx3 +mx2 + nx+ o+ ⟨x5⟩ | k, l,m, n, o ∈ Z℘}. Further, we write Z℘[x]/⟨x5⟩ as Z℘[x]/⟨x5⟩ = {kx4 + lx3 +mx2 + nx+ o | k, l,m, n, o ∈ Z℘}. The graph Γ(Z℘[x]/⟨x5⟩) has ℘4−1 vertices and 1 2(4 ℘5−5℘4−℘2+2) edges. The structure of this graph was originally defined by Musyarrofah et al. in [24]. The following expression represents the vertex set of Γ(Z℘[x]/⟨x5⟩). A = {kx4 + lx3 +mx2 + nx | k, l,m ∈ Z℘, n ∈ Z℘ \ {0̄}}, |A| = ℘4 − ℘3, B = {kx4 + lx3 +mx2 | k, l ∈ Z℘,m ∈ Z℘ \ {0̄}}, |B| = ℘3 − ℘2, C = {kx4 + lx3 | k ∈ Z℘, l ∈ Z℘ \ {0̄}}, |C| = ℘2 − ℘, D = {kx4 | k ∈ Z℘ \ {0̄}}, |D| = ℘− 1. The adjacency matrix of Γ(Z℘[x]/⟨x5⟩) is provided by the following lemma Lemma 2. [24] Adjacency matrix of graph G ∼= Γ(Z℘[x]/⟨x5⟩) is A(G) =  A B C D A O℘4−℘3 O(℘4−℘3)×(℘3−℘2) O(℘4−℘3)×(℘2−℘) N(℘4−℘3)×(℘−1) B O(℘3−℘2)×(℘4−℘3) O℘3−℘2 N(℘3−℘2)×(℘2−℘) N(℘3−℘2)×(℘−1) C O(℘2−℘)×(℘4−℘3) N(℘2−℘)×(℘3−℘2) N℘2−℘ − I℘2−℘ N(℘2−℘)×(℘−1) D N(℘−1)×(℘4−℘3) N(℘−1)×(℘3−℘2) N(℘−1)×(℘2−℘) N℘−1 − I℘−1 . Here O represents the zero matrix, N represents the matrix of ones, and I represents the identity matrix. V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 6 of 19 The eigenvalues of Γ(Z℘[x]/⟨x5⟩) are provided by the following theorem. Theorem 1. Let G ∼= Γ(Z℘[x]/⟨x5⟩). Then the following hold for G. (i) The eigenvalues of G are 0, with multiplicity ℘4 − ℘2 − 2, and −1, with multiplicity ℘2 − 3. (ii) The other eigenvalues of G are solutions to the following polynomial λ4 − ( ℘2 − 3 ) λ3 − ( 2℘5 − 3℘4 + 2℘2 − 2 ) λ2 + ℘3 ( ℘3 − ℘2 − 2℘− 1 ) (℘− 1)2λ+ ℘6(℘− 1)4 = 0. Proof. Suppose that λ be the eigenvalues of G ∼= Γ(Z℘[x]/⟨x5⟩). Based on Lemma 2 has been obtained adjacency matrix of a graph G, then the matrix A(G) − λI can be expressed as: A(G)− λI =  C(−λ, 0,℘4−℘3) O(℘4−℘3)×(℘3−℘2) O(℘4−℘3)×(℘2−℘) N(℘4−℘3)×(℘−1) O(℘3−℘2)×(℘4−℘3) C(−λ, 0,℘3−℘2) N(℘3−℘2)×(℘2−℘) N(℘3−℘2)×(℘−1) O(℘2−℘)×(℘4−℘3) N(℘2−℘)×(℘3−℘2) C(−λ, 1,℘2−℘) N(℘2−℘)×(℘−1) N(℘−1)×(℘4−℘3) N(℘−1)×(℘3−℘2) N(℘−1)×(℘2−℘) C(−λ, 1,℘−1)  = [ P Q R S ] . If λ ̸= 0, then P is invertible and according to Lemma 1, the determinant of A(G)− λI is given by det(A(G)− λI) = det(P ) · det(S −RP−1Q). Since P is a diagonal matrix, its determinant is det(P ) = (−λ)℘ 4−℘2 . (2) Moreover, because P can be written as circulant matrix C(−λ, 0,℘4−℘2), its inverse can be computed using Proposition 2, that is P−1 = 1 (−λ)℘ 4−℘2C ( (−λ) ℘4 −℘2 −1, 0,℘4−℘2 ) = (−λ)℘ 4−℘2−1 (−λ)℘ 4−℘2 I℘4−℘2 = − 1 λ I℘4−℘2 , Substituting this inverse into RP−1Q, we find RP−1Q = − 1 λ [ (℘3 − ℘2)N℘2−℘ (℘3 − ℘2)N(℘2−℘)×(℘−1) (℘3 − ℘2)N(℘−1)×(℘2−℘) (℘4 − ℘2)N℘−1 ] , V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 7 of 19 Thus, S −RP−1Q becomes S −RP−1Q = C(−λ+ b λ ,1+ b λ ,c) ( 1 + b λ ) Nc×d( 1 + b λ ) Nd×c C(−λ+ a λ ,1+ a λ ,d)  = [ X1 Y Y T X2 ] , where a = ℘4 − ℘2, b = ℘3 − ℘2, c = ℘2 − ℘, dan d = ℘− 1. Applying Lemma 1 once again, det(S −RP−1Q) = det(X1) · det(X2 − Y TX−1 1 Y ). By Proposition 1, it can be seen that det(X1) = (−λ− 1)c−1 f(λ), (3) where f(λ) = −λ2+(c−1)λ+cb λ . Also by Proposition 2, X−1 1 = 1 (−λ− 1)c−1 f(λ) C(φc−1, ϑc−1, c), where φc−1 = ( −λ+ c− 2 + cb−b λ ) (−λ− 1)c−2 and ϑc−1 = ( −1− b λ ) (−λ− 1)c−2. Let g(λ) = −λ+ c− 2 + cb− b λ , and h(λ) = −1− b λ . Then φc−1 = (−λ− 1)c−2g(λ) and ϑc−1 = (−λ− 1)c−2h(λ). Thus, X−1 1 = 1 (−λ− 1)f(λ) C(g(λ), h(λ), c). Also, X2 − Y TX−1 1 Y = C( −λ+ a λ − c(b+λ)2 λ2f(λ) , 1+ a λ − c(b+λ)2 λ2f(λ) , d ). By Proposition 1, the determinant of X2 − Y TX−1 1 Y is det ( X2 − Y TX−1 1 Y ) = ( −λ+ a λ − c(b+ λ)2 λ2f(λ) + (d− 1) ( 1 + a λ − c(b+ λ)2 λ2f(λ) )) (−λ− 1)d−1 V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 8 of 19 = ( −λ3f(λ) + (d− 1)λ2f(λ) + daλf(λ)− dc(b+ λ)2 λ2f(λ) ) (−λ− 1)d−1 = ϕ(λ) λ2f(λ) (−λ− 1)d−1. (4) where ϕ(λ) is ϕ(λ) = λ4 − ( ℘2 − 3 ) λ3 − ( 2℘5 − 3℘4 + 2℘2 − 2 ) λ2 + ℘3 ( ℘3 − ℘2 − 2℘− 1 ) (℘− 1)2λ+ ℘6(℘− 1)4. Based on equation (2), (3), and (4), we obtain determinant A(G)− λI, is det(A(G)− λI) = det(P ) · det(X1) · det ( X2 − Y TX−1 1 Y ) = (−λ)℘ 4−℘2−2 · (−λ− 1) ℘2−3 · ϕ(λ). Thus the characteristic polynomial of A(G) is (−λ)℘ 4−℘2−2 · (−λ− 1) ℘2−3 · ϕ(λ) = 0, Hence, 0 and 1 are eigenvalue of G with multiplicity ℘4−℘2−2 and ℘2−3 respectively. The other eigenvalues of G are solutions to the following polynomial λ4 − ( ℘2 − 3 ) λ3 − ( 2℘5 − 3℘4 + 2℘2 − 2 ) λ2 + ℘3 ( ℘3 − ℘2 − 2℘− 1 ) (℘− 1)2λ + ℘6(℘− 1)4 = 0. Theorem 2. Let G ∼= Γ(Z℘[x]/⟨x5⟩). Then En(G) ≥ ℘2 − 3 + √ 4℘5 − 5℘4 − 2℘2 + 5 + 12(℘6(℘− 1)4)1/2 and En(G) ≤ ℘2 − 3 + 2 √ 4℘5 − 5℘4 − 2℘2 + 5. Proof. Let ζ1, ζ2, ζ3, ζ4 be an eigenvalues that satisfy λ4 − ( ℘2 − 3 ) λ3 − ( 2℘5 − 3℘4 + 2℘2 − 2 ) λ2 + ℘3 ( ℘3 − ℘2 − 2℘− 1 ) (℘− 1)2λ+ ℘6(℘− 1)4 = 0. (5) It is obtained that the sum and the product of the eigenvalues from the equation (5) are V.H. Krisnawati, N. Hidayat, A.F. Musyarrofah / Eur. J. Pure Appl. Math, 18 (2) (2025), 5971 9 of 19 as follows 4∑ i=1 ζi = ℘2 − 3,∑ 1≤i