Matrix Analysis of Hexagonal Model and Its Applications in Global Mean-First-Passage Time of Random Walks

Recent advances in graph-structured learning have demonstrated promising results on the graph classification task. However, making them scalable on huge graphs with millions of nodes and edges remains challenging due to their high temporal complexity. In this paper, by the decomposition theorem of Laplacian polynomial and characteristic polynomial we established an explicit closed-form formula of the global mean-first-passage time (GMFPT) for hexagonal model. Our method is based on the concept of GMFPT, which represents the expected values when the walk begins at the vertex. GMFPT is a crucial metric for estimating transport speed for random walks on complex networks. Through extensive matrix analysis, we show that, obtaining GMFPT via spectrums provides an easy calculation in terms of large networks.

text organization, predicting chemical venomousness, and categorizing public buildings in human interactions. Though the permuting indices and the encoding's runtime effectiveness are hurdles in graph classification, for a simple and less order graphs, it is easy to construct an adjacency matrix to check the properties of such graphs [14]. As a result, the best encoding strategy for simple and finite graph classification is indices over node permutations. The adjacency matrix strategy is also convenient in neural graphs to the limitations and worldwide locations of the set of nodes [32]. Prevailing graph classification methods frequently necessitate an adjacent assessment of the structures or rely entirely on algebraic and spectral symbol, which are difficult to calculate. Appropriate illustration approaches are mandatory to encrypt the atomic assembly of the display concisely, which is well-organized. However, transformation invariance, scalability, in addition the programming's runtime proficiency are hurdles in graph classification. Because of the deficiency in order of graph vertices, numerous adjacency matrices can represent the same graph. Consequently, VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the optimum programming method for graph classification is the best invariant further down transformations of the vertices. The current network classification methods frequently require an adjacent assessment of the structures or rely entirely on arranged algebraic spectral investigation, both of which are difficult to investigate. It has recently been used to quantify the robustness of networks in distributed networked control systems based on noisy data. In reality, it has many comparable descriptions, such as the spectrums of graphs and it can be used to extract graph representations. It is renowned that hexagonal systems play a significant part in theoretical chemistry, since they are usual graph symbols of benzenoid hydrocarbon [11]. As a result, hexagonal systems have received a lot of attention. Kennedy and Quintas investigated the enumerations on perfect matchings in an arbitrary hexagonal chain model [17]. In [10] and [22], the authors determined the Wiener index (resp. Edge-Szeged index) of a hexagonal model. Li et al. [33] studied the normalized Laplacian of a penta-graphene with applications. For further studies on laplacian and normalized laplacian we refer [34], [35]. The reference [18] provided a comprehensive explanation of the distinctive polynomial of a hexagonal model. In [28], an explicit closed-form formula for the sum of a resistance distances of hexagonal chain is obtained with the help of Laplacian spectrum.

II. PRELIMINARIES
The networks in this paper are simple, undirected, finite and connected. Let N = (U N , E N ) be a network, where U N denotes the node set, and E N its links respectively. We denote the order of N as n = |U N | and its size as |E N |. For further notations we referred to [1], [4], [24], and [25].
Let A(N ) denotes an adjacency matrix of N , where the entry (i, j) contains 1 if and only if ij ∈ E N and 0, otherwise. Define the Laplacian matrix of N as (N ) = D(N ) − A(N ). We assume that µ 1 <µ 2 ⩽ · · · ⩽ µ n be the spectrums of (N ). It is obvious that if and only if N is a connected network, then µ 1 = 0 and µ 2 > 0. For further studies on (N ), we refer to the following interesting papers [13], [20], [21]  The fact that λ 1 = 0 is well known assumption in spectral graph theory, and λ 2 > 0 when the graph G is assumed to be connected. We denote the spectral of (G) with Sp(G) = {λ 1 , λ 2 , . . . , λ n }. For more details on (G), we suggest [13].
For distance, among the nodes i, j of a graph G is defined as the dimension of a through i-j path in G [5].
Let denotes the characteristic polynomial of any n × n matrix then ϕ( ) = det(tI − ). The matrix I is called a unitary matrix having of .
The automorphism of any network G is defined as a permutation of the nodes in G that maps links to links. Let be an automorphism in any network G; thereby, one may define it as the product of transpositions and disjoint 1-cycles, that is, = (t 1 )(t 2 ) · · · (t m )(l 1 , q 1 )(l 2 , q 2 ) · · · (l k , q k ).
Following Lemmas are well known matrix-tree theorem. Lemma 2.2 ( [16], [19]): Let G be an n-vertex connected [26]): The cycle is denoted by C n and having n vertices, then R ij (C n ) = n 3 −n

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Rendering the considered vertices of a HM n , as described in Fig. 1, an automorphism of HM n is given as g = Thus, ξ 11 and ξ 12 are constructed as follows: Assume that the spectrums of a matrix S are µ j , j = 1, 2, . . . , 2n. Note that µ j > 0 for all j. In view of Lemma 2.1, one has the Laplacian spectrum of T n is where α 2n ̸ = 0.

III. THE MFPT OF HM n AND IMPORTANT LEMMAS
Given a graph G, the MFPT F ij of any node j is the smallest number of steps of any random walk requires to reach at point j. The (MFPT) F ij is defined as the expected value of F j once walk starts at vertex i. MFPT is a vital quantity which is supposed to be useful to approximate the speed of any transport of the random walks of any graphs [15], [31]. The GMFPT denoted by ⟨F(G)⟩ actions the distribution competence of any walk, and obtained by averaging F ij over (|V G |−1) probable end point and |V G | roots of elements [12], that is with the fact that |V G | ̸ = 1. By [9], commuting time C ij among the vertices i and j are accurately 2|E G |r ij , i.e., Lemma 3.1: Assume that B be a 2n × 2n matrix given below: be two submatrices of B. Assume that η i := det χ i and η ′ i := det χ ′ i . We fix η 0 = 1, η ′ 0 = 1. Then for 0 ⩽ i ⩽ 2n, one has and VOLUME 11, 2023 Proof: First, we show (5). To check that η 1 = 4, η 2 = 7, η 3 = 24 is straightforward. In case 3 ⩽ i ⩽ 2n, we expand det χ i with regards to its last row In case, 0 ⩽ i ⩽ n, assume that c i = η 2i and for 0 ⩽ i ⩽ n−1, let d i = η 2i+1 and c 0 = 1, d 0 = 4. In case i ⩾ 1, we have From the first equation in (7), one has Substituting the values of d i−1 and d i in the second part of (7) gives Thus, η i fulfill the following recurrence Then the characteristic equation of (8) is r 4 = 6r 2 − 1, and its roots are . Hence, general solution of (8) is given by Together with the IC,s in (8) give the followings The unique solution of this system can be found to be . We get our result by putting ζ 1 , ζ 2 , ζ 3 and ζ 4 in (9). Through, the parallel directions as above, it is straightforward to obtain (6), which is omitted here.
Then we claim the coefficient of r 2n−1 in F(r) is the same as the coefficient of r 2n−1 in µ 1 (r)µ 2 (r). In fact, as desired. Similarly, the coefficient of r 2n−3 in F(r) is the same as the coefficient of r 2n−3 in µ 1 (r)µ 2 (r). Hence, in order to determine −α 2n−1 , it suffices for us to determine the coefficients of r 2n−1 and r 2n−3 in F(r). □ By a direct calculation, we have .
Notice that 1 + √ 2 and 1 − √ 2 are the roots of r 2 − 2r − 1. Hence, assume that where a, b, c, d belongs to the real numbers. Linking both sides of (13) gives a = −c = 3 16 √ 2 and b = d = 0, that is Hence, coefficient of r 2n−1 in .
Through a parallel directions, we get that the coefficient of ) 2n+1 . Thus, the coefficient of VOLUME 11, 2023 Bear in mind that Then by a parallel discussion as above we can get the coefficients of r 2n−1 in Through the same directions, we establish the coefficient of By a direct calculation, Hence, in view of (3.5), one has Proof: Note that |V HM n | = 4n. By Lemma 2.2, one has here θ i (1 ⩽ i ⩽ 2n − 1) and µ j (1 ⩽ j ⩽ 2n) represents the spectrums of the matrices R and S . On the one hand, in view of Lemma 2.2 we have On the other hand, µ 1 , µ 2 , . . . , µ 2n are the roots of det(rI 2n − S ) = r 2n + α 1 r 2n−1 + · · · + α 2n−1 r + α 2n = 0.
By Vieta's Theorem, one has 2n Together with (16) and (17), our result follows immediately. □ In order to obtain R ij (HM n ), it suffices to determine α 2n−1 and det S in (15). Based on Lemmas 3.1-3.5, we obtain Lemma 3.6: Let HM n be a zig-zag polyhex nanotube with n hexagons. Then

IV. PROOF OF THEOREM
Note that |E HM n | = 5n and |V HM n | = 4n. From (3) and (4), the GMFPT for HM n is Hence, we get our desired result.

V. NUMERICAL CONSEQUENCES AND DISCUSSIONS
In this section, by using Matlab, we give some graphical interpretations between the number of hexagons (n) and ⟨F(HM n )⟩. We also investigated the effect of ⟨F(HM n )⟩ for n = 3 and 4. For the sake of simplicity, we assume ⟨F(HM n )⟩ = M g . In Fig. 2, it shows that M g increases as we increase the hexagons (n). In Fig. 3, it shows that M g increases for both n = 3 and n = 4, but the slope of M g for n = 4 is larger than for n = 3. This means that the GMFPT works efficiently for the large nodes. Hence, we developed a unified strategy for obtaining the scaling properties of ⟨F(HM n )⟩  and achieved an organized study for GMFPT. Since, the study of spectrums are crucial in determining the scaling of GMFPT. Therefore, we used a closed-form formula for GMFPT and all pairs of nodes. Finally, looking at GMFPT in a network, we found that as the number of hexagons grows, so does GMFPT. This demonstrates that hexagons and network invariants have a direct relationship. The GMFPT between source and target exhibits search efficiency when we analyze many random walks equally. We demonstrated in Fig.2 and Fig.3 that GMFPT works efficiently for the large nodes.

VI. CONCLUDING REMARKS
In this contribution, we obtained the GMFPT of HM n . Note that Carmona, Encinas and Mitjana studied the resistance distances for ladder-like graphs [7]. Very recently, Barrett, Evans and Francis [2] studied the effective resistances in straight linear 2-trees (i.e., linear triangle chain) and some related problems. It is quite motivating to study the effective resistances for the Möbius hexagonal ring, and the Möbius pentagonal ring. We will do it in the near future.

Declarations Conflicts of interest/Competing interests:
The authors declare no any conflict of interest/competing interests.
Data availability: Not applicable Code availability: Not applicable Authors' contributions: (All author contributed equally to this work.)