The Kemeny’s Constant and Spanning Trees of Hexagonal Ring Network

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Introduction
Obtaining the total number of spanning trees of any network is the central part of exploration in network theory, as spanning trees of any network grow exponentially through a network size.Earlier in the 1960s, researchers around the world explored numerous procedures of fluctuating efficiency methods.It uses various fields of computer science such as image processing, networking, and countless other usages of minimum spanning trees or entirely possible spanning trees of a network.
Another network invariant is entitled Kemeny's constant (Ω).In [1], the Kemeny's constant is proposed by Kemeny and spell.It is motivating to perceive that this unique network invariant is closely related to the analogous Spectrum of the normalized Laplacian (see Lemma 2.2 in the next section).Kemeny's constant is formally defined as the expected number of steps desirable for the transition from a starting node to a terminus node.It is chosen randomly by a stationary distribution of unbiased random walks on network N.In finite ergodic Markov chains, the Ω has an essential CMC, 2022, vol.73, no.3 property independent of the initial state of the Markov chain [2].The adjacency matrix A(N) of N is a matrix whose (i, j)-entry is 1 if and only if ij ∈ E N and 0, otherwise.Define the Laplacian matrix of N as L(N) = D(N) − A(N), where D(N) is the diagonal matrix whose main diagonal entries are the degrees in N. In recent years, the method of using eigenvalues of normalized Laplacian, Γ(N), consisting of the matrix in spectral geometry and random walks [3,4], attracted the researchers due to its numerous applications.

Preliminaries
All the networks considered in this paper are finite, connected, simple, and undirected.Let N = (U N , E N ) be any network, where U N denote the node-set and E N denote the link set.We represent the order of N as n = |U N | and its size as |E N |.The traditional notation and terminology not defined in this paper are referred to [2,3].
The adjacency matrix A(N) of N is a matrix whose (i, j)-entry is 1 if and only if ij ∈ E N and 0, otherwise.Define the Laplacian matrix of N as L(N) = D(N) − A(N), where D(N) is the diagonal matrix whose main diagonal entries are the degrees in N. We assume that μ 1 < μ 2 • • • μ n be the eigenvalues of L(N).It is obvious that μ 1 = 0 and μ 2 > 0 if and only if N is a connected network.Further, regarding the results on L(N), we recommend the recent work [4] and the references within.
Let M be an m×n matrix.We assume that S ⊂ {1, 2, . . ., m} and T ⊂ {1, 2, . . ., n}.Denote M(S|T) for the submatrix of M, which is obtained by deleting the rows of S and the columns of T. Notably, we denote M(S|T) by M(i|j), where S = {i} and T = {j}.
In recent years, the method using eigenvalues of normalized Laplacian, (N), which consists of the matrix in spectral geometry and random walks [5,6], has attracted more and more researchers' attention.Defining the normalized Laplacian of nonregular networks also attracted researchers.Furthermore, the normalized Laplacian of any network is defined as: Here, when a degree of the node w j in N is 0, then (d j ) − 1 2 = 0, see [5].That is to say 0, otherwise, The notation (Γ(N)) ij symbolizes the (i, j)-entry of Γ(N), and we assume that {λ 1 , λ 2 , . . ., λ n } denote the Spectrum of the normalized Laplacian of N.These eigenvalues are labeled as 0 = λ 1 < λ 2 • • • λ n , with the fact that N is connected if and only if λ 2 > 0. In [7], Chen and Zhang determined that the resistance distance can also be obtained from eigenvalues expressions and their multiplicities in the sense of normalized Laplacian.
The hexagonal system plays an essential role in theoretical chemistry.Since the hexagonal systems are natural network illustrations of benzenoid hydrocarbon [8].Therefore, in various fields, hexagonal systems have been widely studied.The perfect matching in random hexagonal chain network is established by Kennedy et al. [9] in 1991.The hexagonal chain for Wiener index and Edge-Szeged index is determined in [10] and [11], respectively.In [12], Lou and Huang gave complete descriptions of the characteristic polynomial of a hexagonal system.
In this paper, motivated by [13][14][15][16][17] and from the normalized Laplacian decomposition theorem, we obtained the explicit closed-form formulations for Ω and τ for n as well as ∇ n .

Definition and Structures of the Two Hexagonal Ring Networks
We denote the linear hexagonal chain with n hexagons by M n .The hexagonal ring network is denoted by n and computed from M n by identifying the opposite boundary links in an ordered way.The Möbius hexagonal ring network ∇ n obtained by M n by identifying the opposite boundary links in a reversed way.
In this paper, we focus on two interesting molecular network types: the hexagonal ring network (see Fig. 1) and the Möbius hexagonal ring network (see Fig. 2).The hexagonal ring network n is the network obtained from the linear hexagonal chain M n by identifying node 1 with (2n + 1) the node 1 with (2n + 1) , respectively.Similarly, the Möbius hexagonal ring network ∇ n is the network obtained from the linear hexagonal chain M n by identifying the node 1 with (2n + 1) , the node 1 with (2n + 1), respectively.In this section, we discuss some vital block matrices, characteristic polynomial, and the automorphisms of N, which will be used to prove our main results.We denote ϕ(B) = det(xI − B) for the characteristic polynomial of a matrix B, where B is a square matrix and I is the corresponding identity matrix.The automorphism of any network N is a permutation π of the nodes of N having the property that uv is a link in the network N, whenever π(u)π(v) is a link in N. We suppose that the network N is an automorphism in π.Therefore, we write it as a 1-cycle of disjoint product and CMC, 2022, vol.73, no.3 its transpositions in the form π Thus, it is easy to compute that |U N | = m + 2k, and assume that U 0 = {1, 2, . . ., m}, U 1 = {1, 2, . . ., k} and U 2 = {1 , 2 , . . ., k }.After an appropriate organization of the nodes in N, the normalized Laplacian matrix Γ(N) can be arranged in the following way The submatrix Γ U ij is formed by rows corresponding to nodes of the network N in U i and columns corresponding to those in U j , where i = 0, 1, 2 and j = 0, 1, 2. We assume that is a block matrix in which the dimension of a block is the same as the corresponding blocks of Γ(N).Note that the automorphism of N is denoted by π.Hence Γ U 11 = Γ U 22 .Considering the unitary transformation TΓ(N)T T yields where In [15], the first author of this article mentioned the decomposition theorem of the Laplacian polynomial.In the following lemma, it is easy to see that the decomposition theorem for normalized Laplacian polynomial is also existed as: Lemma 2.1: The matrices (N), R (N) and S (N) as defined above are, satisfies that ϕ( The following two lemmas are essential to obtain our main results.

Lemma 2.2:
[18] The Kemeny's constant of a simple connected network N with n nodes is denoted by and defined as Lemma 2.3: [5] The Spanning trees of a network N with order n and links m are denoted by τ (N) and defined as τ 3 Important Matrices and the Spectrum of ( n ) According to Lemma 2.1, we firstly obtain the normalized Laplacian eigenvalues for n .Then we give the formula for the sum of the normalized Laplacian eigenvalues' reciprocals and the product of the normalized Laplacian eigenvalues, which motivate us to calculate the Ω and the number of spanning trees of n .We also deduce the corresponding results based on our achieved results.Bearing in mind the labeled nodes of n as shown in Fig. 1, one can see that π is an automorphism of the network n .That is to say, U 0 = ∅, U 1 = {1, 2, . . ., 2n} and U 2 = {1 , 2 , . . ., (2n) }.From the notation in (1), we may denote Γ R ( n ) and Γ S ( n ) as Γ R and S respectively, and we have The matrices Γ U 11 and Γ U 12 are of order 2n × 2n as given below: CMC, 2022, vol.73, no.3 and For the sake of simplicity, we denote eigenvalues of Γ R and Γ S are respectively, as Further on, we introduce a matrix named Q, where Q is a matrix constructed from Γ R with the (1, 2n)-entry and the (2n, 1)-entry by replacing 0. We consider the i-th order principal submatrix, Q i (resp.C i ), formed by the first i rows and corresponding columns (resp.the last i rows and corresponding columns) of Q.Put q i := det Q i and c i := det C i .Put q 0 = 1, c 0 = 1 and it is straightforward that q i = c i for all even i.Lemma 3.1: For 0 i 2n, Proof.Since it is easy to see that q 1 = 2 3 , q 2 = 1 2 , q 3 = 2 9 .For 3 i 2n, expanding det Q i with respect to its last row yields ( From the first equation in (2), one has e i−1 Hence, e i = d i+1 + 1 6 d i .Substituting e i−1 and e i into the second equation in (2) Keeping the same procedure, one can obtain that e i+1 = 1 3 e i − 1 36 e i−1 , i 1, and q i satisfies the below recurrence relation Then, the characteristic equation of ( 3) is x 4 = 1 3 x 2 − 1 36 , the roots of which are The general solution of (3) is given by Together with the initial conditions of (4), the system of equations yields The unique solution of this system can be found to be , , , . We get our desired result by substituting y 1 , y 2 , y 3 and y 4 in (4).
Considering the procedure as the proof of Lemma 3.1, it is easy to determine the following results.

Lemma 3.4:
is the sum of all those principal minors of Γ R which have 2n − 2 rows and columns (see [19, P5]), one has We proceed further by considering the below subcases.
Together with the convention that det Q = 1, whence j = 2. Then where together with the convention that det Notice that for even j, Otherwise, it is evident that det Hence, we introduce a matrix F, where F is a matrix obtained from Γ S with the (1, 2n)-entry and the (2n, 1)-entry by replacing 0. We give the detail for i-th order of principal submatrix, F i (resp.U i ), obtain from the first i rows and corresponding columns (resp.the last i rows and corresponding columns) of F. Put f i := det F i , u i := det U i and fixed f 0 = 1, u 0 = 1.Through the below observations, we proceed further.

Observation 3.5:
Proof of Observation 3.5: expanding det F i respect to its last row yields For i 1, we set that s 0 = 1 and t 0 = 4 3 , then one has From the first equation of (5), we have Putting the values of t i−1 and t i into the second equation in (5) gives By keeping the same procedure, we have t i+1 = t i − 1 36 t i−1 , i 1.Therefore, f i satisfies the below recurrence relation CMC, 2022, vol.73, no.3 Proof of Observation 3.8: Since −h 2n−1 (= (−1) 2n−1 h 2n−1 ) is the sum of all those principal minors of Γ S which have 2n − 1 rows and columns (see also in [19, P5]), one has −h 2n−1 = 2n i=1 det S (i|i).For 2 i 2n − 1, one has .
By the same procedure as in the detail of det Γ R (i|i)(2 i 2n − 1) in Lemma 3.3, we have This completes the proof of Observation 3.8.
The below proposition is a direct consequence of Lemma 2.2.
Proposition 3.9: Let n be a zig-zag polyhex network with n hexagons.Then The eigenvalues of Γ R are characterized as 0 = η 1 < η 2 • • • η 2n and the eigenvalues of In the following propositions, we derived the expressions where a 2n−1 = 0. Then η 2 , η 3 , . . ., η 2n are the roots of the following equation From the Vieta's Theorem, one has 2n i=2 where h 2n = 0. Then ξ 1 , ξ 2 , . . ., ξ 2n are the roots of the following equation Bear in mind the Vieta's Theorem; we have 2n i=1 To obtain the expression 2n i=1 , it is enough to obtain h 2n−1 and det Γ S in (10).In view of (10), Observations 3.7 and 3.8, Proposition 3.11 follows directly.Now, we will calculate some significant invariants related to n (resp.∇ n ) by the expression of the eigenvalues of Γ(N).We also contribute closed-form formulae of Ω and τ for n (resp.∇ n ) in the subsequent section.
Together with Lemma 2.3, we get our desired result.

The Kemeny's Constant and the Number of Spanning Trees of ∇ n
Now, we devote our attention to determining the Ω and the τ for the Möbius hexagonal ring network n .Based on the labeled nodes of ∇ n as shown in Fig. 2, one has π = (1, 1 )(2, 2 ) • • • (2n, (2n) ) is an automorphism of ∇ n .From Fig. 2, we have U 0 = ∅, U 1 = {1, 2, . . ., 2n} and U 2 = {1 , 2 , . . ., (2n) }.For the sake of simplicity, we denote R (∇ n ) and S (∇ n ) to R and S , respectively.Thereby, it is easy to see that R = R and Note that η 1 , η 2 , . . ., η 2n are the spectrums of R and suppose that β j (1 i 2n) are the spectra of S .Due to Lemma 2.1, we have the normalized Laplacian eigenvalues of ∇ n is {η 1 , . . ., η 2n , β 1 , . . ., β 2n }.In the following theorem, we give the formula for Ω and the τ for ∇ n .In this paper, bear in mind the spectrums of normalized Laplacian; we identified the explicit closed-form formulae of Ω and τ for n and ∇ n , respectively.It is natural and exciting to study the hitting times of random walk for the hexagonal ring network and the Möbius hexagonal ring network.We will do it shortly.

Funding Statement:
The authors received no specific funding for this study.

Conflicts of Interest:
The authors declare they have no conflicts of interest to report regarding the present study.

Figure 1 :
Figure 1: The hexagonal ring network

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Based on the relationship between roots and coefficients of ϕ(Γ R ) and ϕ(Γ S ) Proposition 3.10: Let

Figure 4 :
Figure 4: Spanning trees and n in n