Normalized Laplacian Energy and Distance Based Energy for Cross Monic Zero Divisor Graphs Associated with Commutative Ring

: Cross monic zero divisor graph for a commutative ring R is a connected graph, denoted by C M Z G ( Z n × Z m [ x ] / ⟨ f ( x ) ⟩ ) with order ξ , whose vertices are non-zero zero divisors Z ( R ) / { 0 } of commutative ring, and two vertices u , v are connected by an edge if and only if uv = 0 . In this paper, we discuss energy, Laplacian energy, distance energy and distance signless Laplacian energy for C M Z G ( Z 2 × Z p > 2 [ x ] / ⟨ x 2 ⟩ ) and C M Z G ( Z p × Z p [ x ] / ⟨ x 2 ⟩ ) . Also, we determine the normalized Laplacian energy.


Introduction
Let R be a commutative ring with multiplicative identity 1 ̸ = 0.If there exists x 2 ∈ R (x 2 ̸ = 0) such that x 1 x 2 = 0 for some x 1 ∈ R (x 1 ̸ = 0), then x 1 is referred to as a zero divisor of R. The collection of zero divisors is symbolized by Z(R), while Z(R)/{0} = Z(R) * is the collection of nonzero zero divisors of R. The zero divisor graph Γ(R) of R is a graph, where Z(R) is its node set and two different nodes y, z ∈ Z(R) are connected if yz = zy = 0. Beck [1] established such graphs over commutative rings in his concept, he incorporated the identity and was primarily concerned with the coloring of a commutative ring.Following that, Anderson et al. [2] updated the concept of Γ(R) by omitting the identity of R. The finite field of order n is represented by F n and a ring of integers modulo n by Z n .The order of Γ(Z n ) is n − 1 − ϕ (n), where as ϕ is Euler's phi function.The graph theoretic characteristics of Γ(Z n ) are widely investigated [3][4][5].Shang [6] focuses on the commutativity aspects within prime near-rings, providing valuable insights that enrich the broader understanding of ring theory.Investigation of the spectral properties of matrices associated with graphs is always interesting and challenging.We note that the graphs associated with different algebraic structures, for instance, power graphs [7], annihilator monic prime graph [8] and commuting graphs of groups [9,10] have helped to solve several problems both in algebra and combinatorics.Alali et al. [11], implies a study of algebraic structures within Z n and their connections with topological indices and entropies, underscoring the interdisciplinary intersection of algebra and graph theory.The adjacency matrix of G is the n × n matrix A = (a i j ), where a i j = 1 if there is an edge between vertex i and vertex j, otherwise a i j = 0.For an n-vertex graph G with adjacency matrix A having eigenvalues This energy encapsulates essential information about the graph's structural properties and connectivity.Specifically, the eigenvalues serve as indicators of the graph's spectral characteristics, offering insights into the algebraic features of the underlying ring.Adjacency eigenvalues of zerodivisor graphs discussed by Young [12].In addition to this, the Laplacian energy graph is ascertained through the eigenvalues of its Laplacian matrix.This Laplacian energy imparts supplementary insights by concentrating on the connections between vertices, capturing the inherent algebraic structure of the graph.Investigating energy and Laplacian energy in zero divisor graphs entails a detailed examination of spectral properties, adding to a more profound comprehension of how algebraic structures and graph-theoretic characteristics interact within this specific context.The distance energy of a graph is denoted by E D and a quantitative measure that reflects the structural properties of the graph based on distances between its vertices.It is defined as the sum of the absolute values of the eigenvalues of the distance matrix of the graph.The distance matrix represents the pairwise distances between vertices in the graph.Rather et al [13], probably extensively explores Laplacian eigenvalues and their ramifications for the zero divisor graph in the domain of modular arithmetic.Similarly, distance energy [14] and distance Laplacian energy (distance signless Laplacian energy) [15,16], linked to the distance Laplacian matrix (distance signless Laplacian matrix) respectively, focus on capturing the relationships between vertices while incorporating distance information (Let . Alhevaz et al. [17] discusses distance signless Laplacian Estrada index combines distance information with the graph's signless Laplacian matrix, offering a comprehensive perspective on the graph's structure, ultimately contributing to the advancement of graph theory and its applications in various fields.This note aims to explore the implications of these distance based energy measures in the context of zero divisor graphs, shedding light on their applications and significance in algebraic graph theory.The normalized Laplacian energy is computed from the eigenvalues of this matrix and serves as a measure of the graph's, how efficiently information can propagate through the networks.Research on normalized Laplacian energy explores its applications in diverse fields, including computer science, physics, and biology.Entries of the normalized Laplacian matrix are some recent work on the normalized Laplacian see [18][19][20]. Motivated by the above articles, we investigate Laplacian energy, distance based energy, and normalized Laplacian energy for cross monic zero divisor graph of a commutative ring.Cross monic zero divisor graph of a commutative ring, denoted C M Z G (Z n × Z m [x]/⟨ f (x)⟩), whose vertices are the non-zero zero divisors of the commutative ring, and whose two vertices u, v are connected by an edge if and only if uv = 0.For example, cross monic zero divisor graph of 1 and Figure 2. Characteristic polynomial and eigenvalues of adjacency matrix, Laplacian matrix, distance matrix, distance Laplacian matrix and distance signless Laplacian matrix of Figure 2 is shown in Table 1 and Table 2.
The structure of this paper is outlined as follows: In Section 2, we explore the energy and Laplacian energy of cross monic zero divisor graphs within the commutative rings Section 3 is dedicated to the examination of the distance energy and distance signless Laplacian energy of cross monic zero divisor graphs.Furthermore, in Section 4, we delve into the discussion of the normalized Laplacian eigenvalues and their energy in the context of cross monic zero divisor graphs.

Energy and Laplacian energy of cross monic zero divisor graphs of commutative ring
Theorem 1 Energy of cross monic zero divisor graph of commutative ring where p is prime number greater than 2.
Proof.Let the cross monic zero divisor graph of Z 2 × Z p [x]/⟨x 2 ⟩ be a simple graph, then the adjacent matrix is where p is odd prime.
Proof.Let the cross monic zero divisor graph of Z p × Z p [x]/⟨x 2 ⟩ be a simple graph with then adjacent matrix is Then, as is well known, we have (see [21], for example).Moreover, since must hold (for example, see [22]), we have Using this together with the Cauchy-Schwartz inequality, applied to the vectors ) and (1, 1, 1, . . ., 1) with 2p 2 − p − 2 entries, we obtain the inequality Thus, we must have Now, since the function f (y) = y + (2p 2 − p − 2)(4p 3 − 7p 2 + p + 2 − y 2 ) decreases on the interval and hence f (λ 1 ) ≤ f ( ) must hold as well.From this fact, and inequality holds.Hence the proof.
Example 1 For cross monic zero divisor graph of commutative ring Z 3 × Z 3 [x]/⟨x 2 ⟩ with order 14 and size 25, we have Solution We consider cross monic zero divisor graph of commutative ring Z 3 × Z 3 [x]/⟨x 2 ⟩ (Figure 3), The adjacent matrix is

Volume 5 Issue 2|2024| 2033
Contemporary Mathematics Then the characteristic polynomial is λ 14 − 25λ 12 − 12λ 11 + 108λ 10 + 96λ 9 = 0.The spectrum of the graph is Theorem 3 Let the cross monic zero divisor graph of where where Laplacian matrix and spectrum of cross monic zero divisor graph of respectively.

Distance based energy of cross monic zero divisor graphs
Theorem 4 Upper and lower bounds of distance energy of cross monic zero divisor graph of

Proof. Distance matrix and spectrum of the graph is
where Proof.Let us choose s a = α 2 a , for a = 1, 2, 3, . . ., ξ .We obtain where Hence, we get the required bounds.Distance Laplacian matrix and Distance signless Laplacian matrix of cross monic zero divisor graph is shown in Table 3.
Table 3. Block matrix of distance (signless Laplacian) of cross monic zero divisor graph Theorem 6 If C M Z G is a connected graph with order ξ and diameter β , then Proof.Since dis ab ≥ 1 for a ̸ = b and there are ξ (ξ − 1) 2 pairs of vertices in C M Z G , then we get Again, dis ab ≤ β for a ̸ = b and there are ξ (ξ − 1) 2 pairs of vertices in C M Z G , then we get

O I
where Proof.Let δ 1 = 0, and hence , therefore Theorem 9 Let Z α × Z β [x]/⟨x 2 ⟩ be connected, C M Z G with smallest δ s and largest δ ξ non-negative normalized Laplacian eigenvalues.Then Proof.Consider ξ > 2, recall the Ozekis inequality [23], stating that p k and q k , 1 ≤ k ≤ ξ , are positive real numbers, then where N 1 = max 1≤k≤ξ p k , N 2 = max 1≤k≤ξ q k , n 1 = min 1≤k≤ξ p k and n 2 = min 1≤k≤ξ q k .An application of Ozekis inequality with p k = 1 and q k = δ k , 2 ≤ k ≤ ξ , yields In view of [24], it is easy to see that yielding the assertion (i).To prove assertion (ii), we recall the Polya-Szego inequality, stating that if p k , q k , N k , n k , 1 ≤ k ≤ ξ are as in part (i), then we have Applying the last inequality p k = 1 and q k = δ k , 2 ≤ k ≤ ξ , we get Therefore

Conclusion
The primary emphasis of this paper is the exploration of Laplacian energy, distance based energy, and normalized Laplacian energy concerning the cross monic zero divisor graph within a commutative ring, denoted as C M Z G (Z n × Z m [x]/⟨x 2 ⟩).The paper also includes visual representations of the concepts discussed.In essence, the contribution of this paper lies in enhancing our comprehension of the graph properties linked to the cross monic zero divisor graph within the framework of commutative ring.

Figure 3 .
Figure 3. C M Z G (Z 3 × Z 3 [x]/⟨x 2 ⟩) be respectively, the distance Laplacian matrix and the distance signless Laplacian matrix, where Diag(Tr) is diagonal matrix of vertex transmissions.Eigenvalues of D L (G) and D Q (G) denoted by ∂ L i and