Spectral graph theory is an important part of algebraic graph theory which mainly utilizes matrix theory, polynomial theory, and combinatorial methods to study the different spectra (or ranges of values) that come from graphs. Furthermore, spectral theory investigates how these spectra are connected to the structure and properties of graphs. It links the algebraic (math-based) aspects of graphs to their topological (shape-based) aspects. By studying how a graph's spectrum relates to its structure, we can not only better understand these graphs but also find useful applications in areas like improving networks, designing computer circuits, and solving operational problems. Some key areas of study in spectral theory include the adjacency spectrum, Laplacian spectrum, signless-Laplacian spectrum, and distance spectrum of graphs. Among these, the Laplacian spectrum is the most studied and produces the most results. Studying the Laplacian spectrum is not only valuable for theoretical knowledge but also has many uses in chemistry, physics, complex networks, and electronic engineering. Over the last few decades, significant attention has been given to examining the structure of graphs, along with their spectral, topological, and combinatorial characteristics. Laplacian eigenvalues have proven useful in identifying various graph invariants, including the Kirchhoff index, global mean-first passage time, and the count of spanning trees. Typically, the characteristic polynomial and spectrum of the graph matrix for certain graph operations, such as the complement, union, Cartesian product, direct product, and strong product, can be derived from the factor graphs. Since complex molecular graphs can be effectively described using graph operations, it is feasible to represent the properties of these complex molecular graphs through the invariants of their factor graphs.

From the perspective of spectral graph theory, numerous structural features and dynamic behaviors of graphs have been investigated. The literature on spectral graph theory covers diverse aspects of Laplacian matrices across different graph structures. Merris provides a comprehensive survey, discussing the Laplacian matrix's spectrum, algebraic connectivity, and applications in areas such as chemistry [1]. Hong and Zhang investigate Laplacian eigenvalues in simple and bipartite graphs, establishing key bounds and structural insights, particularly for trees and regular graphs [2]. Agaev and Chebotarev (2006) extended the study to weighted directed graphs, exploring the connections between Laplacian and stochastic matrices and their semiconvergent properties [3]. Rojo and Soto focused on unweighted rooted trees, analyzing the eigenvalues of adjacency and Laplacian matrices based on symmetric tridiagonal matrices [4]. Ding and Jiang investigate the spectral norms and eigenvalue distributions of random graph matrices, revealing key convergence behaviors aligned with Wigner's semi-circular law [5]. Wu extends the use of Laplacian matrices into quantum mechanics, exploring conditions for separability in weighted graphs [6]. Kaveh and Rahami (2006) focus on the eigenvalues and eigenvectors of graph products, presenting efficient methods for solving eigenproblems in structural mechanics, especially for Cartesian and lexicographic products [7]. Spielman emphasizes the significance of Laplacian matrices in algorithm design, highlighting their role in fast solutions for linear equations and their application to graph theory through innovations like graph sparsifiers and local clustering [8].

In 2012, Estrada introduced path Laplacian matrices as a new concept that generalized the combinatorial Laplacian and applied them to consensus analysis in networks, thus showing its potential for enhancing network synchronization and other applications [9]. In their 2013 publication, Krishnan et al. came up with a novel scheme of multi-level preconditioning for Laplacian matrices used in computer graphics which resulted in substantial performance benefits in applications such as image colorization and mesh processing [10]. Another similar work by Dong et al. aimed toward exploring laplacian matrix learning for smooth graph signal representation that contributed to advancement in graph signal processing [11]. Pirani and Sundaram analyzed the smallest eigenvalue properties of grounded Laplacian matrices giving out insights into spectral graph theory and its applications [12]. Efficient methods have been developed by Bergamaschi and Martínez to approximate the generalized inverse of Laplacian matrices which are important when solving large scale graph problems [13]. Recent research on Laplacian matrices has led to notable advancements. In 2016, Jog and Kotambari analyzed the spectra of coalesced complete graphs, studying the adjacency, Laplacian, and signless Laplacian energies to understand their spectral properties and applications [19]. Moving to 2018, Bandeira explored random Laplacian matrices, revealing that the largest eigenvalue often approximates the largest diagonal entry, with implications for convex relaxation techniques and Erdos-Rényi graph connectivity thresholds [14]. Li provided insights into the constrained Rayleigh quotient for eigen-balanced Laplacian matrices, which proved valuable for cooperative control problems and convergence rates in consensus protocols [16]. The work by Bergamaschi and Bozzo focused on comparing algorithms for computing the smallest eigenpairs of graph Laplacians, including the Implicitly Restarted Lanczos Method and Jacobi-Davidson method, particularly for large, sparse networks [18]. Zhou et al. introduced an optimal neighborhood multi-view spectral clustering algorithm, which enhances clustering performance by effectively combining first-order and high-order Laplacian matrices [15]. Hermann and Konigorski addressed the optimization of edge weights in directed graph Laplacians to achieve desired spectral properties [17].

In 2019, Alhevaz et al. explored the Brouwer-type conjecture related to the eigenvalues of the distance signless Laplacian matrix. Their findings provided bounds for the sums of the largest and smallest eigenvalues, applying these results to graphs with specific diameters and transmission properties [20]. Moving forward to 2022, Ganie and Shang investigated the spectral radius and energy of the signless Laplacian matrix of digraphs, proposing new lower bounds and characterizing extremal digraphs based on vertex degrees and walk lengths [21]. Also in 2022, Morbidi examined matrix functions of the Laplacian matrix and their applications to distributed formation control, showing how these functions can enhance performance and flexibility in consensus protocols [22]. Recent studies have significantly advanced our understanding of various Laplacian matrices and their applications. In 2021, Reinhart introduced the normalized distance Laplacian matrix, offering new insights into its spectral properties and connections with the normalized Laplacian matrix. The study showed that this matrix has fewer cospectral pairs compared to other matrices [23]. The same year, Chakrabarty et al. explored the spectral properties of adjacency and Laplacian matrices in inhomogeneous Erdõs-Rényi random graphs. Their work detailed the empirical spectral distributions and their convergence to deterministic limits [24].

The paper by Alazemi et al. [25] explores chain graphs, a specific class of bipartite graphs, with unique Laplacian eigenvalues. The authors provide structural insights, degree constraints, and analyze the eigenspaces of these graphs. Notably, they highlight conditions such as the absence of vertex triplets sharing identical neighborhoods and propose applications in Laplacian dynamics, including the controllability of multi-agent systems. Meanwhile, Anđelić et al. [26] introduce a family of tridiagonal matrices with eigenvalues as perfect squares, applying this result to analyze the Laplacian controllability of half graphs, a subclass of chain graphs, further advancing the understanding of spectral graph theory. The authors in [28] investigate the Laplacian controllability of graphs formed using standard graph products, including joins, Cartesian, tensor, and strong products. The study provides theoretical insights and introduces an iterative method to construct infinite families of controllable Laplacian pairs. Additionally, Anđelić et al. [29] focus on the Q-index, the largest eigenvalue of the signless Laplacian matrix, for connected graphs with fixed order and size. The authors derive spectral bounds for the Q-index of nested split graphs, offering both theoretical results and computational comparisons to improve understanding of spectral properties in graph theory.

More recently, in 2023, Bapat et al. extended the concept of bipartite matrices by examining the bipartite Laplacian matrix of nonsingular trees. They provided a combinatorial description of this matrix and established several key identities [27]. Additionally, Mallik expanded the Matrix Tree Theorem to signed graphs, introducing a new oriented incidence matrix and offering a combinatorial formula for the determinant of the signless net Laplacian matrix [30]. Raza et al. focused on generalized prism graphs and found that spectral analysis helps measure network features like passage time and path length [31]. Raza and Munir extended this by showing how Laplacian and signless Laplacian spectra can be used to understand network properties and predict behaviors in various fields [32]. In a later study, Raza et al. applied these methods to torus grid graphs, deriving key network measures and improving our knowledge of network structures [33].

We define a path of length α_m ∈ ℕ as the graph P_αm that has vertex set V = {v ∈ ℕ : 0 ≤ v ≤ α_m} and where two vertices determine an edge if and only if |v_i−v_j| = 1 for v_i, v_j ∈ V. Then, a mesh network graph Mmn, often known as two-dimensional lattice graph or grid graph, is defined as the Cartesian product Mmn=Pαm⊡Pαn, exhibits a total of 2mn−m−n = (m−1)n+(n−1)m edges, reflecting the combined count of horizontal and vertical edges. Simultaneously, Mmn boasts mn vertices, aligning with the Cartesian product of the vertex sets of P_m and P_n. The mentioned graph operation have gained significant attention in the field of graph theory and computer science. These graphs are widely used to model spatial relationships and connectivity in various applications, such as computer networks, image processing, and computational geometry. The study of grid graphs has evolved over the years, with researchers exploring their properties, algorithms, and applications. Harel and Sardashti [34] presented a comprehensive analysis of the structural characteristics of mesh network graph, highlighting their regularity and symmetry. Smith et al. [35] investigated efficient algorithms for computing shortest paths in mesh network graphs, providing valuable insights into optimizing navigation in grid-based environments. Additionally, Chen and Du [36] explored the application of Mmn in wireless sensor networks, showcasing their relevance in practical scenarios. Recent work by Hinz and Holz auf der Heide [35] delved into the dynamic aspects, addressing challenges related to real-time updates and adaptability. Furthermore, the survey by Kumar et al. [37] offers a holistic overview of mesh network graph applications and algorithmic advancements. Building on earlier research about mesh network graphs and their spectra, our study thoroughly examined the Adjacency et al. Laplacian spectra of Mmn. We didn't just calculate these spectra; we also applied them to real-world network analysis. Using our results, we computed important network measures such as graph energies, Kirchhoff index, mean-first passage time, path length, spanning trees, and spectral radius. This approach aimed to give a deeper insight into the properties of Mmn, such as its connectivity, resilience, and efficiency. In this section, we review key findings from earlier studies that relate to the solutions discussed in this paper. This study distinguishes itself by conducting a comprehensive spectral analysis of generalized mesh networks, focusing on the adjacency, Laplacian, and signless Laplacian spectra to derive explicit expressions for critical network parameters such as the Kirchhoff index, spectral radius, average path length, global mean first passage time, graph energies, and the number of spanning trees. While previous research has applied spectral methods to various network types, such as torus networks and categorical product networks, your work uniquely emphasizes generalized mesh networks, providing detailed spectral characterizations that enhance the understanding of their structural and dynamic properties. The complete structure of our article is presented hierarchically in Figure 1. By presenting results graphically, your study offers clear visualizations of how these parameters vary with network dimensions, facilitating deeper insights into their interplay and impact. This approach not only broadens the applicability of spectral methods but also offers a robust framework for exploring and optimizing complex real-world networks, thereby contributing valuable perspectives across multiple scientific disciplines.

Before discussing graph-based matrices and the related lemmas associated with the Kronecker product, let's first revisit the notion of ψ-sum graphs. Let $ = (V($), E($)) be a simple undirected graph, where V($) represents its vertex set and E($) represents its edge set. The number of vertices in $ is denoted by |V($)|, and the number of edges is denoted by |E($)|. If an edge e connects two vertices u and v, the edge uv can also be referred to as e. For a given vertex v ∈ V($), its neighborhood in $, denoted by N_$(v), is the set of vertices adjacent to v, specifically N_$(v) = {u ∈ V($)∣uv ∈ E($)}. The degree of a vertex v, symbolized as d_$(v), is the number of vertices in its neighborhood, i.e., d_$(v) = |N_$(v)|.

Definition 1. The diagonal matrix is defined as Dg($)=diag[dυij] for i = j and the Laplacian matrix Lp($)=Dg($)-Ad($) is defined by the subtraction of the adjacency matrix from the diagonal matrix of vertex degrees. Elaborating in matrix form, Lp($) is defined as:

Lemma 1. Let E ∈ M_{p, p}(G), F ∈ M_{p, q}(G), G ∈ M_{q, p}(G), H ∈ M_{q, q}(G) with H being invertible, such that

Then,

Lemma 2. Let C = (c_ij) ∈ M_{r, s}(G), D ∈ M_{t, u}(G). Then the Kronecker product of C and D is defined as

Lemma 3. Let C and D be square matrices of order r and s, respectively, with eigenvalues λ_i (1 ≤ i ≤ r) and ν_j (1 ≤ j ≤ s). Then the eigenvalues of C⊗I_s+I_r⊗D are λ_i+ν_j. Moreover, if V_i is an eigenvector of C corresponding to λ_i and W_j is an eigenvector of D corresponding to ν_j, then V_i⊗W_j is an eigenvector of C⊗I_s+I_r⊗D corresponding to λ_i+ν_j.

Lemma 4. Let C ∈ M_{r, s}(G), D ∈ M_{t, u}(G), E ∈ M_{r, t}(G), F ∈ M_{s, u}(G), and β ∈ G. The following properties hold:

(a) (C⊗D)^T = C^T⊗D^T.

(b) (C⊗D)(E⊗F) = (CE)⊗(DF).

(c) (C⊗D)⊗E = C⊗(D⊗E).

(d) β(C⊗D) = βC⊗D = C⊗βD.

(e) If C and D are invertible, then (C⊗D)⁻¹ = C⁻¹⊗D⁻¹.

Lemma 5. The eigenvalues of the Adjacency matrix, Laplacian matrix, and Signless Laplacian matrix for a path graph Pn are expressed as 2cos(πkn+1), 2-2cos(πkn), and 2+2cos(2πkn+1), respectively, where k = 0, 1, 2, …, n−1.

In this section, we have evaluated the exact values for the Adjacency, Laplacian and signless Laplacian spectrum of the generalized Mesh Network graphs utilizing the graph and algebra techniques. Theorem 1 provides expressions for the sum of reciprocals and the product of adjacency eigenvalues of the generalized mesh graph Mmn, which are fundamental in understanding network connectivity and robustness [38]. The sum of the reciprocals of the eigenvalues is often associated with resistance distance and other network invariants, while their product is related to graph determinant properties, which have applications in quantum networks and structural analysis [39]. Extending this analysis, Theorem 2 focuses on the Laplacian eigenvalues, which play a crucial role in describing network dynamics such as diffusion processes and synchronization [41]. The sum of the reciprocals of the Laplacian eigenvalues is connected to important network measures like Kirchhoff's index, influencing resistance-based properties, while their product is associated with the number of spanning trees, a key quantity in evaluating network reliability and resilience [42]. Furthermore, Theorem 3 explores the Signless Laplacian spectrum, which is particularly useful in applications involving directed flows and energy distribution in networks [40]. The sum of the reciprocals of these eigenvalues helps in analyzing clustering tendencies in complex networks, whereas their product provides a measure of structural stability and modular properties. These interpretations establish strong connections between spectral properties and real-world applications, enhancing the accessibility of the results for researchers in diverse fields such as physics, computer science, and engineering.

Theorem 1. Let the summation of the reciprocals and the product of the adjacency eigenvalues of the generalized mesh graph Mmn be denoted by Zmn and Wmn, respectively. Then:

Proof. The adjacency matrix of the mesh graph Mmn is:

By matrix addition, it can be expressed as:

Using Lemma 2, we have:

The matrix

is the adjacency matrix of Q_m, a path graph with m vertices. Thus:

Now, assume two invertible matrices U and V related to the matrices Q_n and Q_m, such that:

and

Since, the eigenvalues of the Adjacency matrix for a path graph Q_n are given by λk=2cos(πkn+1), k=0,1,2,…,n-1, so the diagonal entries of the upper triangular matrices are:

Consequently:

and the diagonal entries of this upper triangular matrix are given by:

Thus, the adjacency eigenvalues for the generalized mesh graph are:

Using this result, we obtain:

and

Corollary 1. For a mesh graph with equal dimensions (n = m), the product and sum of the reciprocals of the adjacency eigenvalues are given by:

The proof follows directly from Theorem 1.

Theorem 2. Let ZnmL and WnmL denote the sum of the reciprocals and the product of the Laplacian eigenvalues, respectively, for the generalized mesh network graph Mmn. Then, these quantities are given by:

and

Proof. The Laplacian matrix associated with the mesh network graph Mmn is expressed as:

By decomposing this matrix, it can be rewritten as:

Referring to Lemma 2, this is further simplified as:

The matrix

is the Laplacian matrix for the path graph Q_n with n nodes. Therefore:

Suppose U and V are invertible matrices related to the matrices Q_n and Q_m, respectively. Then:

Since, The eigenvalues of the Laplacian matrix for a path graph Q_n are given by μk=2-2cos(πkn), k=0,1,2,…,n-1, so the diagonal entries of the upper triangular matrices are:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

Clearly:

with diagonal elements of the resulting matrix given by:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

Thus, the eigenvalues of the Laplacian matrix for the mesh network graph are:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

Finally, using the eigenvalues in Equation 1, we derive:

and

Corollary 2. For a mesh network graph of equal dimensions (n = m), the product and reciprocal of the sum of the Laplacian eigenvalues are given by:

The proof follows directly from Theorem 2.

Theorem 3. Let the sum of the reciprocals and the product of all Signless Laplacian eigenvalues of the generalized mesh graph Mmn be denoted by SnmQ and PnmQ, respectively. Then,

Proof. The Signless Laplacian matrix for the mesh graph Mmn is expressed as:

which can be broken down by matrix addition as follows:

According to Lemma 1.1, we have:

The matrix given by

is, in fact, the Signless Laplacian matrix of the path graph P_n with n vertices. Thus, we obtain:

Introducing two invertible matrices A and B that correspond to the matrices P_n and P_m, we have:

and

Since, the eigenvalues of the Signless Laplacian matrix for a path graph Q_n are given by qk=2+2cos(2πkn+1), k=0,1,2,…,n-1 so the diagonal entries of the upper triangular matrices are:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

Clearly, the following holds:

where the diagonal elements of this matrix are given by:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

Thus, the adjacency eigenvalues for the mesh graph can be expressed as:

where i = 0, 1, 2, …, m−1 and j = 0, 1, 2, …, n−1.

From the results in Equation 3, we can derive:

and

Corollary 3. For regular dimension mesh graph (n = m), the products and reciprocals of the sums of Signless Laplacian eigenvalues are defined as

The proof is obvious by Theorem 3.

The framework developed in the previous section allows for the calculation of important network metrics, including graph energy, Kirchhoff index KI, spectral radius SR, average path length APL, global mean first passage time MFT, and the number of spanning trees TN. To facilitate these computations, two key quantities, MA and MB, are introduced. The quantity MA is defined as the product of all non-zero eigenvalues, denoted as λ_i, of a given matrix, while MB is the sum of the reciprocals of these eigenvalues:

Here, λ_i represents the eigenvalues of the Laplacian matrix associated with the graph Mmn, where i ranges from 1 to n. These quantities serve as the basis for further analysis and provide a deeper understanding of various network properties.

In this section, we developed a MATLAB Algorithm 1 with a total run time of 0.182 seconds to produce Tables 1, 2. These tables provide exact values for several key metrics: Kirchhoff index KI, Spectral radius SR, Average path length APL, Global mean first passage time MFT, Graph energies AE, and the number of spanning trees TN. The algorithm is designed for the generalized mesh network graph Mmn. In Table 1, k is set to 2, while p varies from 2 to 15. For Table 2, k is fixed at 3. Exact values for these metrics are calculated to provide a detailed understanding of the network's behavior across different dimensions. In addition to the tables, Figure 2 visually represents the relationships between network size and variations in KI, SR, APL, MFT, AE, and TN, further improving the interpretation of the results.

A key observation in the graphical representations is the clear trend indicating that as the network expands, several key metrics increase significantly. These visuals enhance the understanding of the network's behavior, complementing the numerical data and offering a more intuitive grasp of the dynamics within the generalized mesh network. The graphical depiction of the results provides a glimpse into the potential of our methodologies. Researchers are encouraged to utilize our carefully developed algorithm and analysis framework to explore the complexities of more sophisticated real-world networks. The flexibility of our approach offers a valuable toolset, enabling a deeper understanding of network behavior and performance in various contexts. This work paves the way for further studies, serving as a platform for future exploration of complex networks with improved precision and efficiency (Algorithm 2).

Table 1 evaluates several network-related parameters for the Generalized Mesh Network Graph M2n across values of n from 2 to 15. The parameters included are the Spectral Radius (SPR), Average Edge Length (AεR), Knot Number (KNI), Average Path Length (APL), and Global Mean First Passage Time (GMPTF). As n increases, a notable trend is observed across these parameters. The Spectral Radius (SPR) shows a slight increase from 4.013 to 5.963, reflecting a gradual growth in the network's connectivity as more nodes are added. The Average Edge Length (AεR) also increases consistently, indicating that as the network grows, the average distance between connected nodes becomes larger. The Knot Number (KNI), which quantifies the number of key nodes in the network, increases significantly, suggesting that more nodes are becoming central as the network expands. The Average Path Length (AP_L) increases from 0.819 to 21.634, showing that the average distance between any two nodes grows with the size of the network. Finally, the Global Mean First Passage Time (GMPTF) increases from 0.8356 to 4.1385, reflecting that it takes more time on average for a random walker to reach a target node as the network becomes larger. Table 1 provides similar parameters for the Generalized Mesh Network Graph M3n. The parameters assessed are the Spectral Radius (SPR), Average Path Length (AP_L), Average Edge Length (AεR), Global Mean First Passage Time (GMPTF), and Knot Number (KNI). Trends in these parameters show a clear pattern of growth and increase with respect to the network size.