Enhanced dung beetle optimizer for Kriging-assisted time-varying reliability analysis

Yunhan Ling; Yiqing Shi; Huimin Hou; Lidong Pan; Hao Chen; Peixin Liang; Shiyuan Yang; Peng Nie; Jiahao Han; Debiao Meng; Yunhan Ling; Yiqing Shi; Huimin Hou; Lidong Pan; Hao Chen; Peixin Liang; Shiyuan Yang; Peng Nie; Jiahao Han; Debiao Meng

doi:10.3934/math.20241420

AIMS Mathematics

2024, Volume 9, Issue 10: 29296-29332. doi: 10.3934/math.20241420

Previous Article Next Article

Research article Special Issues

Enhanced dung beetle optimizer for Kriging-assisted time-varying reliability analysis

1.
China Academy of Machinery Beijing Research Institute of Mechanical & Electrical Technology Co., Ltd., Beijing 100083, China
2.
INEGI, Faculty of Engineering, University of Porto, Porto, 4200-465, Portugal
3.
School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China
4.
Institute of Electronic and Information Engineering of UESTC in Guangdong, Dongguan 523808, China

Received: 11 August 2024 Revised: 13 September 2024 Accepted: 19 September 2024 Published: 16 October 2024
MSC : 60K10, 62N05, 90C23

During the engineering structure's operation, the mechanical structure's performance and loading will change with time, so the parameter uncertainty and structural reliability will also have dynamic characteristics. The time-varying reliability analysis method can more accurately evaluate structural reliability by fully using this dynamic uncertainty. However, the time-varying reliability analysis was mainly based on the spanning rate method, which was complex and difficult to obtain the final result. Therefore, this study proposed an enhanced dung beetle optimization (EDBO) assisted time-varying reliability analysis method based on the adaptive Kriging model. With the help of the adaptive Kriging model and the EDBO optimization algorithm, the efficiency of the time-varying reliability analysis method was improved. At the same time, to prevent prematurely falling into the local search trap, the method improved the uniformity of the sample by initializing the sample through improved tent chaotic mapping (ITCM). Next, the Gaussian random walk strategy was used to search the updated position, which further improved the accuracy of the reliability analysis results. Finally, the accuracy and effectiveness of the proposed time-varying reliability analysis method were verified by four mechanical structure model examples. From the calculation results, it can be seen that with the help of the new DBO optimization algorithm, the relative error of the proposed reliability analysis results was about 20%~30% lower than that of the traditional reliability analysis method. What's more, the calculation efficiency was higher than that of other reliability analysis methods.

Keywords:

Citation: Yunhan Ling, Yiqing Shi, Huimin Hou, Lidong Pan, Hao Chen, Peixin Liang, Shiyuan Yang, Peng Nie, Jiahao Han, Debiao Meng. Enhanced dung beetle optimizer for Kriging-assisted time-varying reliability analysis[J]. AIMS Mathematics, 2024, 9(10): 29296-29332. doi: 10.3934/math.20241420

Related Papers:

[1]	Zhi-Yu Shi, Jia-Bao Liu . Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum. AIMS Mathematics, 2024, 9(3): 5431-5450. doi: 10.3934/math.2024262
[2]	Jahfar T K, Chithra A V . Central vertex join and central edge join of two graphs. AIMS Mathematics, 2020, 5(6): 7214-7233. doi: 10.3934/math.2020461
[3]	Zenan Du, Lihua You, Hechao Liu, Yufei Huang . The Sombor index and coindex of two-trees. AIMS Mathematics, 2023, 8(8): 18982-18994. doi: 10.3934/math.2023967
[4]	Jia-Bao Liu, Kang Wang . The multiplicative degree-Kirchhoff index and complexity of a class of linear networks. AIMS Mathematics, 2024, 9(3): 7111-7130. doi: 10.3934/math.2024347
[5]	Akbar Ali, Sadia Noureen, Akhlaq A. Bhatti, Abeer M. Albalahi . On optimal molecular trees with respect to Sombor indices. AIMS Mathematics, 2023, 8(3): 5369-5390. doi: 10.3934/math.2023270
[6]	Kun Wang, Wenjie Ning, Yuheng Song . Extremal values of the modified Sombor index in trees. AIMS Mathematics, 2025, 10(5): 12092-12103. doi: 10.3934/math.2025548
[7]	Zhen Lin, Ting Zhou, Xiaojing Wang, Lianying Miao . The general Albertson irregularity index of graphs. AIMS Mathematics, 2022, 7(1): 25-38. doi: 10.3934/math.2022002
[8]	Zhibin Du, Ayu Ameliatul Shahilah Ahmad Jamri, Roslan Hasni, Doost Ali Mojdeh . Maximal first Zagreb index of trees with given Roman domination number. AIMS Mathematics, 2022, 7(7): 11801-11812. doi: 10.3934/math.2022658
[9]	Tariq A. Alraqad, Igor Ž. Milovanović, Hicham Saber, Akbar Ali, Jaya P. Mazorodze, Adel A. Attiya . Minimum atom-bond sum-connectivity index of trees with a fixed order and/or number of pendent vertices. AIMS Mathematics, 2024, 9(2): 3707-3721. doi: 10.3934/math.2024182
[10]	Zhenhua Su, Zikai Tang, Hanyuan Deng . Higher-order Randić index and isomorphism of double starlike trees. AIMS Mathematics, 2023, 8(12): 31186-31197. doi: 10.3934/math.20231596

Abstract

1. Introduction

Throughout this article, we only consider simple, undirected, and finite graphs and assume that all graphs are connected. Suppose $\mathscr{G}$ is a graph with the vertex set $V_(\mathscr{G}) = \{v_{1}, \; v_{2}, \cdot \cdot \cdot, \; v_{n}\}$ and the edge set $E_(\mathscr{G}) = \{e_{1}, \; e_{2}, \cdot \cdot \cdot, \; e_{m}\}$ . The adjacency matrix $A(\mathscr{G})$ is a $0-1 n \times n$ matrix indexed by the vertices of $\mathscr{G}$ and defined by $a_ij = 1$ if and only if $v_{s}v_{t} \in E_{\mathscr{G}}$ . For more notation, one refer to ^[1].

The Laplacian matrix of graph $\mathscr{G}$ is defined as $L(\mathscr{G}) = D(\mathscr{G}) - A(\mathscr{G})$ , and assume that the eigenvalues of $L(\mathscr{G})$ are labeled $0 \leq \mu_{1} < \mu_{2} < \cdot \cdot \cdot < \mu_{n}$ .

$\begin{eqnarray} (\mathcal{L}(\mathscr{G}))_{st} = \begin{cases} 1, & s = t;\\ -1, & s\neq{t}\; ,\; v_{s}\sim v_{t}; \\ 0, & otherwise. \end{cases} \end{eqnarray}$

(1.1)

The normalized Laplacian matrix is given by

$\begin{eqnarray} (\mathcal{L}(\mathscr{G}))_{st} = \begin{cases} 1, & s = t;\\ -\frac{1}{\sqrt{d_{s}d_{t}}}, & s\neq{t}\; ,\; v_{s}\sim v_{t}; \\ 0, & otherwise. \end{cases} \end{eqnarray}$

(1.2)

The distance, $d_{ij} = d_{\mathscr{G}}(v_s, v_t)$ , between vertices us and ut of $\mathscr{G}$ is the length of the shortest $u_{s}$ , $u_{t}$ -path in $\mathscr{G}$ . The Wiener index ^[2,3] is the sum of the distances of any two vertices in the graph $\mathscr{G}$ , that is

$\begin{eqnarray*} W(G) = \sum\limits_{s < t}d_{st}. \end{eqnarray*}$

In 1947, the distance-based invariant first appeared in chemistry ^[2,3] and began to the applied to mathematics 30 years later [5]. Nowadays, the Wiener index is widely used in mathematics ^[6,7,8] and chemistry ^[9,10,11].

In a simple graph $\mathscr{G}$ , the degree, $d_{ij} = d_{\mathscr{G}}(v_i)$ , of a vertex $v_{i}$ is the number of edges at $v_{i}$ . The Gutman index ^[12] of the simple graph $\mathscr{G}$ is expressed by

$\begin{eqnarray*} Gut(\mathscr{G}) = \sum\limits_{s < t}d_{s}d_{t}d_{st}. \end{eqnarray*}$

Klein and Randic initially outlined the concepts associated with the resistance distance of the graph. Assume that each edge is replaced by a unit resistor, and we use rst to denote the resistance distance between two vertices s and t. Similar to the Wiener index, the Kirchhoff index ^[13,14] of graph $\mathscr{G}$ is expressed as the sum of the resistance distances between each two vertices, that is

$\begin{eqnarray*} Kf(\mathscr{G}) = \sum\limits_{s < t}r_{st}. \end{eqnarray*}$

In 2007, Chen and Zhang ^[15] defined the multiplicative degree-Kirchhoff index ^[16,17], that is

$\begin{eqnarray*} Kf^{*}(\mathscr{G}) = \sum\limits_{s < t}d_{s}d_{t}r_{st}. \end{eqnarray*}$

Phenyl is a conjugated hydrocarbon, and $L_{n}^{6, 4, 4}$ denote a linear chain, containing $n$ hexagons and $2n-1$ squares, as shown in Figure 1.

Figure 1. Graphs of

$L_{n}^{6, 4, 4}$ and

$L_{n}$ .

DownLoad: Full-Size Img PowerPoint

With the rapid changes of the times, organic chemistry has also developed rapidly, which has led to a growing interest in polycyclic aromatic compounds.

In 1985, the computational method and procedure of the matrix decomposition theorem were proposed by Yang ^[18]. This led to the solution of some problems in graph networks and allowed the unprecedented development of self-homogeneous linear hydrocarbon chains. For example, in 2021, X.L. Ma ^[20] got the normalized Laplacian spectrum of linear phenylene, and the linear phenylene containing it has $n$ hexagons and $n-1$ squares. L. Lan ^[21] explored linear phenylene with $n$ hexagons and $n$ squares. Umar Ali ^[22] analyzed the strong prism of a graph $G$ , which is the strong product of the complete graph of order $2$ , and X.Y. Geng ^[23] obtained the Laplacian spectrum of $L_{n}^{6, 4, 4}$ , which contains $n$ hexagons and $2n-1$ squares. J.B. Liu ^[24] derived the Kirchhoff index and complexity of $O_{n}$ , which denoting linear octagonal-quadrilateral networks. C. Liu ^[25] got the Laplacian spectrum and Kirchhoff index of $L_n$ , and $L_n$ has $t$ hexagons and $3t+1$ quadrangles.

Inspired by these recent works, we try to investigate the Laplacians and the normalized Laplaceians for graph transformations on phenyl dicyclobutadieno derivatives.

The various sections of this article are as follows: In Section 2, we proposed some concepts and lemmas and used them in subsequent articles. In Section 3 and Section 4, we acquired the Laplacian matrix and the normalized Laplacian matrix, then we made sure the Kirchoff index, the multiplicative degree-Kirchoff index, and the complexity of $L_n$ . In Section 5, we obtained conclusions based on the calculations in this paper.

2. Preliminary works

In this article, graph $L_n$ and graph $L_{n}^{6, 4, 4}$ are portrayed in Figure $1$ . Define the characteristic polynomial of matrix $U$ of order $n$ as $P_{U}(x) = det(xI-U)$ .

It is easy to understand that $\pi = (1, 1')(2, 2')\cdots\big((4n), (4n)'\big)$ is an automorphism. Let $V_{1} = \{u_{1}, u_{2}, \cdots, u_{4n+1}, v_{1}, \cdots, v_{4n}\}, \; V_{2} = \{u'_{1}, u'_{2}, \cdots, u'_{4n}, v'_{1}, \cdots, v'_{4n+1}\}$ , $|V(L_{n})| = 8n$ and $|E(L_{n})| = 19n-4$ . Thus, the (normalized) Laplacians matrix can be expressed in the form of a block matrix, that is

$\begin{equation*} \mathcal{L}(L_{n}) = \left( \begin{array}{ccc} \mathcal{L}_{V_{0}V_{0}}& \mathcal{L}_{V_{0}V_{1}}& \mathcal{L}_{V_{0}V_{2}}\\ \mathcal{L}_{V_{1}V_{0}}& \mathcal{L}_{V_{1}V_{1}}& \mathcal{L}_{V_{1}V_{2}}\\ \mathcal{L}_{V_{2}V_{0}}& \mathcal{L}_{V_{2}V_{1}}& \mathcal{L}_{V_{2}V_{2}}\\ \end{array} \right), \end{equation*}$

where $L_{V_{s}V_{t}}$ and $\mathcal{L}_{V_{s}V_{t}}$ are a submatrix consisting of rows corresponding to the vertices in $V_s$ and columns corresponding to the vertices in $V_t$ , $s, t = 0, 1, 2$ . Let

$\begin{equation*} Q = \left( \begin{array}{ccc} I_{t} & 0& 0\\ 0 & \frac{1}{\sqrt{2}}I_{4n}& \frac{1}{\sqrt{2}}I_{4n}\\ 0 & \frac{1}{\sqrt{2}}I_{4n}& -\frac{1}{\sqrt{2}}I_{4n}\\ \end{array} \right), \end{equation*}$

then

$\begin{equation*} Q{L}(L_{\mathscr{G}})Q' = \left( \begin{array}{cc} L_{A}(\mathscr{G})& 0\\ 0& L_{S}(\mathscr{G}) \end{array} \right), Q{L}(\mathcal{L}_{\mathscr{G}})Q' = \left( \begin{array}{cc} \mathcal{L}_{A}(\mathscr{G})& 0\\ 0& \mathcal{L}_{S}(\mathscr{G}) \end{array} \right), \end{equation*}$

and $Q'$ is the transposition of $Q$ .

$L_{A} = L_{V_{1}V_{1}}+L_{V_{1}V_{2}}$ , $L_{S} = L_{V_{1}V_{1}}-L_{V_{1}V_{2}}$ , $\mathcal{L}_{A} = \mathcal{L}_{V_{1}V_{1}}+\mathcal{L}_{V_{1}V_{2}}$ , $\mathcal{L}_{S} = \mathcal{L}_{V_{1}V_{1}}-\mathcal{L}_{V_{1}V_{2}}$

Lemma 2.1. ^[20] If $\mathscr{G}$ is a graph and suppose that $L_{A}(\mathscr{G}), L_{S}(\mathscr{G}), L_{A}(\mathscr{G})$ , and $L_{S}(\mathscr{G})$ are determined as above, then

$\begin{eqnarray*} \vartheta_{L(L_{n})}(y) = \theta_{L_{A}(\mathscr{G})}(y) \theta_{L_{S}(\mathscr{G})}(y), \vartheta_{\mathcal{L}(L_{n})}(y) = \theta_{\mathcal{L}_{A}(\mathscr{G})}(y) \theta_{\mathcal{L}_{S}(\mathscr{G})}(y). \end{eqnarray*}$

Lemma 2.2. ^[26] With the extensive study of the Kirchhoff index, Gutman and Mohar proposed an algorithm based on the relation between the Kirchhoff index and the Laplacian eigenvalues, namely

$\begin{eqnarray*} Kf(\mathscr{G}) = n\sum\limits_{t = 2}^{n}\frac{1}{\xi_{t}}, \end{eqnarray*}$

and the eigenvalues of $L(\mathscr{G})$ are $0 = \xi_{1} < \xi_{2} \geq \cdots \geq \xi_{n}(n\geq2)$ .

Lemma 2.3. ^[14] Suppose that the eigenvalues of $L(\mathscr{G})$ are $\xi_1 \leq \xi_2 \leq \cdots \leq \xi_n$ , then its multiplicative degree-Kirchhoff index can be denoted by

$\begin{eqnarray*} Kf^{*}(\mathscr{G}) = 2m\sum\limits_{t = 2}^{n}\frac{1}{\xi_{t}}. \end{eqnarray*}$

Lemma 2.4. ^[1] The number of spanning trees in $\mathscr{G}$ can also be called the complexity of $\mathscr{G}$ . If $\mathscr{G}$ is a graph with $|V_\mathscr{G}| = n$ and $|E_\mathscr{G}| = m$ . Let $\lambda_i(i = 2, 3, \cdots, n)$ be the eigenvalues of $L(\mathscr{G})$ . Then the complexity of $\mathscr{G}$ is

$\begin{eqnarray*} 2m\tau(\mathscr{G}) = \prod\limits_{i = 1}^{n}d_i\cdot\prod\limits_{i = 2}^{n}\lambda_i. \end{eqnarray*}$

3. Kirchhoff index of $L_n$

In this section, the main objective is to find out the Kirchhoff index of $L_n$ . Then, combining the definition of the Laplacian matrix and Eq (1.1), we can write these block matrices as follows:

$\begin{eqnarray*} L_{V_1 V_1}& = & \left( \begin{array}{cccccccccccccc} 3 & -1 &&&&&&&&&&\\ -1 & 4 &-1 &&&&&&&&& \\ &-1 & 5 &-1 &&&&&&&& \\ &&-1&5&-1&&&&&&& \\ &&&-1&5&-1&&&&&&\\ &&&&-1&4&-1&&&&&\\ &&&&&&\ddots&&&&&\\ &&&&&&-1&5&-1&&&\\ &&&&&&&-1&4&-1&&\\ &&&&&&&&-1&5&-1&\\ &&&&&&&&&&-1&3\\ \end{array} \right)_{(4n)\times(4n)}, \end{eqnarray*}$

and

$\begin{eqnarray*} L_{V_1 V_2}& = & \left( \begin{array}{cccccccccccccc} -1&-1&&&&&&&&&&\\ -1&0&-1 &&&&&&&&& \\ &-1 & -1 &-1 &&&&&&&& \\ &&-1&-1&-1&&&&&&& \\ &&&-1&-1&-1&&&&&&\\ &&&&-1&0&-1&&&&&\\ &&&&&&\ddots&&&&&\\ &&&&&&-1&-1&-1&&&\\ &&&&&&&-1&0&-1&&\\ &&&&&&&&-1&-1&-1&\\ &&&&&&&&&&-1&-1\\ \end{array} \right)_{(4n)\times(4n)}. \end{eqnarray*}$

Hence,

$\begin{eqnarray*} L_{A}& = & \left( \begin{array}{cccccccccccccc} 2&-2&&&&&&&&&&\\ -2&4&-2 &&&&&&&&& \\ &-2 & 4 &-2 &&&&&&&& \\ &&-2&4&-2&&&&&&& \\ &&&-2&4&-2&&&&&&\\ &&&&-2&4&-2&&&&&\\ &&&&&&\ddots&&&&&\\ &&&&&&-2&4&-2&&&\\ &&&&&&&-2&4&-2&&\\ &&&&&&&&-2&4&-2&\\ &&&&&&&&&&-2&2\\ \end{array} \right)_{(4n)\times(4n)}, \end{eqnarray*}$

and

$\begin{equation*} L_{S} = diag(4,4,6,6,6,4, \cdot \cdot \cdot ,6,4,6,4). \end{equation*}$

Assume that $0\leq \alpha_1 < \alpha_2\leq \alpha_3 \leq \cdots \leq \alpha_{4n}$ are the roots of $P_{L_{A}}(x) = 0$ , and $0 \leq \beta_1 \leq \beta_2 \leq \beta_3 \leq \cdots \leq \beta_{4n}$ are the roots of $P_{L_{S}}(x) = 0$ . By Lemma 2.2, we immediately have

$\begin{eqnarray} Kf(L^{2}_{n}) = 2(4n)\Bigg(\sum\limits_{i = 2}^{4n}\frac{1}{\alpha_{i}}+\sum\limits_{j = 1}^{4n}\frac{1}{\beta_{j}}\Bigg). \end{eqnarray}$

(3.1)

Obviously, $\sum_{j = 1}^{4n}\frac{1}{\beta_{j}}$ can be obtained according to $L_{S}$ .

$\begin{eqnarray} \sum\limits_{j = 1}^{4n}\frac{1}{\beta_{j}} = \frac{1}{6}\times(3n-2)+\frac{1}{4}\times(n+2) = \frac{9n+2}{12}. \end{eqnarray}$

(3.2)

Next, we focus on computing $\sum_{i = 1}^{4n}\frac{1}{\alpha_{i}}$ . Let

$\begin{equation*} P_{L_{A}}(x) = det(xI-L_{A}) = x(x^{4n-1}+a_{1}x^{4n-2}+\cdots+a_{4n-2}x+a_{4n-1}), a_{4n-1}\ne 0. \end{equation*}$

Based on Vieta's theorem of $P_{L_{A}}(x)$ , we can exactly get the following equation:

$\begin{eqnarray} \sum\limits_{i = 2}^{4n}\frac{1}{\alpha_{i}} = \frac{(-1)^{4n-2}a_{4n-2}}{(-1)^{4n-1}a_{4n-1}}. \end{eqnarray}$

(3.3)

For the sake of convenience, let $M_s$ be used to express the $s-th$ order principal minors of matrix $A$ , and $m_s = detM_s$ is recorded. We can get $m_1 = 2, m_2 = 4, m_3 = 8$ .

And

$\begin{equation*} m_s = 4m_{s-1}-4m_{s-2}, 4\leq s \leq 4n, \end{equation*}$

by further induction, we have

$\begin{equation*} m_s = 2^{s}. \end{equation*}$

In this way, we can get two theorems.

Theorem 3.1. $(-1)^{4n-1}a_{4n-1} = (4n)2^{4n-1}$ .

Proof. Due to the sum of all the principal minors of order $4n-1$ of $L_A$ is $(-1)^{4n-1}a_{4n-1}$ ,

$\begin{eqnarray*} (-1)^{4n-1}a_{4n-1}& = &\sum\limits_{s = 1}^{4n}det L_{A}[s]\\ & = &\sum\limits_{s = 1}^{4n}det \left( \begin{array}{cc} M_{s-1} &0 \\ 0 & U_{4n-s}\\ \end{array} \right)\\ & = &\sum\limits_{s = 1}^{4n}det M_{s-1}\cdot det U_{4n-s}, \end{eqnarray*}$

where

$\begin{eqnarray*} M_{s-1}& = & \left( \begin{array}{cccc} l_{11} & -2 & \cdots &0\\ -2 & l_{22} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & l_{s-1,s-1}\\ \end{array} \right)_{(s-1)\times(s-1)}, \end{eqnarray*}$

$\begin{eqnarray*} U_{4n-s}& = & \left( \begin{array}{cccc} l_{s+1,s+1} & \cdots & \cdots &0\\ \vdots & \ddots & \vdots & \vdots\\ \vdots & \cdots & l_{4n-1,4n-1} & \vdots\\ 0 & \cdots & -2 & l_{4n,4n}\\ \end{array} \right)_{(4n-s)\times(4n-s)}. \end{eqnarray*}$

Let $m_0 = 1, det U_0 = 1$ , because of the symmetry of matrix $L_A$ , then $det U_{4n-s} = det M_{4n-s}$ . Hence

$\begin{eqnarray*} (-1)^{4n-1}a_{4n-1}& = &\sum\limits_{s = 1}^{4n}det m_{s-1}\cdot det m_{4n-s}\\ & = &(4n)2^{4n-1}, \end{eqnarray*}$

as desired. □

Theorem 3.2. $(-1)^{4n-2}a_{4n-2} = \frac{(4n-1)(4n)(4n+1)2^{4n-1}}{3}$ .

Proof. Since the $(-1)^{4n-2}a_{4n-2}$ is the tatal of all the principal minors of order $4n-2$ of $L_A$ , we have

$\begin{eqnarray*} (-1)^{4n-1}a_{4n-1}& = &\sum\limits_{1\leq s < t \leq 4n}det L_{A}[s,t]\\ & = &\sum\limits_{1\leq s < t \leq 4n}det M_{s-1}det N_{t-s-1}det U_{4n-t}, \end{eqnarray*}$

where

$\begin{eqnarray*} L_{A}[s,t]& = & \left( \begin{array}{ccc} M_{p-1} & 0 & 0\\ 0 & N_{t-s-1} & 0\\ 0 & 0 & U_{4n-t}\\ \end{array} \right),1\leq s < t \leq 4n \end{eqnarray*}$

and

$\begin{eqnarray*} N_{t-s-1}& = & \left| \begin{array}{ccccccc} 4 & -2 & & & & & \\ -2 & 4 & -2 & & & & \\ & -2 &4 & -2 & & & \\ &&&\ddots&&&\\ & & & -2 & 4&-2 &\\ & & & & -2& 4 & -2\\ & & & & &-2 & 4\\ \end{array} \right|_{(t-s-1)}\\ & = &(t-s)2^{t-s-1}. \end{eqnarray*}$

Therefore, we can have

$\begin{eqnarray*} (-1)^{4n-2}a_{4n-2}& = &\sum\limits_{1\leq s < t \leq 4n}det M_{s-1}\cdot det N_{t-s-1}\cdot det U_{4n-t}\\ & = &\sum\limits_{1\leq s < t \leq 4n}(t-s)2^{t-s-1}\cdot det {m_{s-1}\cdot m_{4n-t}}\\ & = &\frac{(4n-1)(4n)(4n+1)2^{4n-1}}{3}. \end{eqnarray*}$

The proof is over. □

From the results of Theorem 3.1 and Theorem 3.2, we can get

$\begin{eqnarray} \sum\limits_{i = 2}^{4n}\frac{1}{\alpha_{i}} = \frac{(-1)^{4n-2}a_{4n-2}}{(-1)^{4n-1}a_{4n-1}} = \frac{16n^2-1}{12}, \end{eqnarray}$

(3.4)

where the eigenvalues of $L_A$ are $0 \leq \alpha_1 < \alpha_2 \leq \alpha_3 \leq \cdots \leq \alpha_{4n}$ .

Lemma 3.3. Suppose $L_{n}^{6, 4, 4}$ be the dicyclobutadieno derivative of phenylenes and the graph $L_n$ be obtained from the transformation of the graph $L_{n}^{6, 4, 4}$ .

$\begin{eqnarray*} Kf(L_n) = \frac{32n^3+18n^2+2n}{3}. \end{eqnarray*}$

Proof. Substituting Eqs (3.5) and (3.6) into (3.4), the Kirchhoff index of $L_n$ can be expressed

$\begin{eqnarray*} Kf(L_n)& = &2(4n)\Bigg(\sum\limits_{i = 2}^{4n}\frac{1}{\alpha_{i}}+\sum\limits_{j = 1}^{4n}\frac{1}{\beta_{j}}\Bigg)\\ & = &(8n)(\frac{9n+2}{12}+\frac{(4n+1)(4n-1)}{12})\\ & = &\frac{32n^3+18n^2+2n}{3}. \end{eqnarray*}$

The result is as desired. □

The Kirchhoff index of $L_n$ from $L_1$ to $L_{12}$ is shown in Table 1.

Table 1. The Kirchhoff indices of

$L_{1}, \; L_{2}, ..., \; L_{12}.$ .

$\mathscr{G}$	$Kf(\mathscr{G})$	$\mathscr{G}$	$Kf(\mathscr{G})$	$\mathscr{G}$	$Kf(\mathscr{G})$
$L_1$	$17.3$	$L_5$	$1486.7$	$L_9$	$8268.0$
$L_2$	$110.7$	$L_6$	$2524.0$	$L_{10}$	$24457.3$
$L_3$	$344.0$	$L_7$	$3957.3$	$L_{11}$	$30454.7$
$L_4$	$781.3$	$L_8$	$5850.7$	$L_{12}$	$37360.0$

| Show Table

DownLoad: CSV

Next, we will further consider the Wiener index of $L_n$ .

Lemma 3.4. Let $L_{n}^{6, 4, 4}$ be the dicyclobutadieno derivative of [n]phenylenes, and the graph $L_n$ be obtained from the transformation of the graph $L_{n}^{6, 4, 4}$ , then

$\begin{eqnarray*} \lim\limits_{n \to \infty}\frac{Kf(L_n)}{W(L_n)} = \frac{1}{4}. \end{eqnarray*}$

Proof. Consider $d_{st}$ for all vertices. For the sake of convenience, we divide the vertices of the graph into the following five categories:

Case 1. Vertex $1$ of $L_n$ :

$\begin{eqnarray*} g_{1}(i) = 1+2(\sum\limits_{k = 1}^{4n-1}k). \end{eqnarray*}$

Case 2. Vertex $4j-3(j = 1, 2, \cdots, n)$ of $L_n$ , $i = 4j-3$ :

$\begin{eqnarray*} g_{2}(i) = 1+2(\sum\limits_{k = 1}^{4n-1}k+\sum\limits_{k = 1}^{4n-i}k). \end{eqnarray*}$

Case 3. Vertex $4j-2(j = 1, 2, \cdots, n)$ of $L_n$ , $i = 4j-2$ :

$\begin{eqnarray*} g_{3}(i) = 1+2(\sum\limits_{k = 1}^{4n-1}k+\sum\limits_{k = 1}^{4n-i}k). \end{eqnarray*}$

Case 4. Vertex $4j-1(j = 1, 2, \cdots, n)$ of $L_n$ , $i = 4j-1$ :

$\begin{eqnarray*} g_{4}(i) = 1+2(\sum\limits_{k = 1}^{4n-1}k+\sum\limits_{k = 1}^{4n-i}k). \end{eqnarray*}$

Case 5. Vertex $4j(j = 1, 2, \cdots, n)$ of $L_n$ , $i = 4j$ :

$\begin{eqnarray*} g_{5}(i) = 1+2(\sum\limits_{k = 1}^{4n-1}k+\sum\limits_{k = 1}^{4n-i}k). \end{eqnarray*}$

Hence, we have

$\begin{eqnarray*} W(L_n)& = &\frac{4g_{1}(i)+2\sum_{i = 4j-3}g_{2}(i)+2\sum_{i = 4j-2}g_{3}(i)+2\sum_{i = 4j-1}g_{4}(i)+2\sum_{i = 4j}g_{5}(i)}{2}\\ & = &\frac{4(1+2\sum_{k = 1}^{4n-1}k)+2\sum_{j = 1}^{n}[1+2(\sum_{k = 1}^{4j-k}k+\sum_{k = 1}^{4n-4j+2}k)]}{2}\\ &+&\frac{2\sum_{j = 1}^{n}[2+2(\sum_{k = 1}^{4j-3}k+\sum_{k = 1}^{4n-4j+2}k)]+2\sum_{j = 1}^{n}[1+2(\sum_{k = 1}^{4j-2}k+\sum_{k = 1}^{4n-4j+1}k)]}{2}\\ &+&\frac{2\sum_{j = 1}^{n-1}[1+2(\sum_{k = 1}^{4j-1}k+\sum_{k = 1}^{4n-4j}k)]}{2}\\ & = &\frac{128n^3+48n^2-5n+3}{3}. \end{eqnarray*}$

Consider the above results of the Kirchhoff index and the Wiener index. We can get following equation when $n$ tends to infinity:

$\begin{eqnarray*} \lim\limits_{n \to \infty}\frac{Kf(L_n)}{W(L_n)} = \frac{1}{4}. \end{eqnarray*}$

The result is as desired. □

4. Multiplicative degree-Kirchhoff index and complexity of $L_n$

In this section, we use the eigenvalues of the normalized Laplacian matrix to determine the multiplicative degree-Kirchhoff index of $L_n$ . Besides, we calculate the complexity of $L_n$ . Then

$\begin{eqnarray*} \mathcal{L}_{V_{1}V_{1}}& = & \left( \begin{array}{ccccccccccccc} 1&\frac{-1}{\sqrt{12}}&&&&&&&&&\\ \frac{-1}{\sqrt{12}}&1&\frac{-1}{\sqrt{20}} &&&&&&&& \\ & \frac{-1}{\sqrt{20}} &1 &\frac{-1}{5}&&&&&&& \\ &&\frac{-1}{5}&1&\frac{-1}{5}&&&&&& \\ &&&\frac{-1}{5}&1&\frac{-1}{\sqrt{20}}&&&&&\\ &&&&\frac{-1}{\sqrt{20}}&1&\frac{-1}{\sqrt{20}}&&&&\\ &&&&&&\ddots&&&&\\ &&&&&&\frac{-1}{5}&1&\frac{-1}{\sqrt{20}}&&\\ &&&&&&&\frac{-1}{\sqrt{20}}&1&\frac{-1}{\sqrt{20}}&\\ &&&&&&&&\frac{-1}{\sqrt{20}}&1&\frac{-1}{\sqrt{15}}\\ &&&&&&&&&\frac{-1}{\sqrt{15}}&1\\ \end{array} \right)_{(4n)\times(4n)}, \end{eqnarray*}$

and

$\begin{eqnarray*} \mathcal{L}_{V_{1}V_{2}}& = & \left( \begin{array}{ccccccccccccc} \frac{-1}{3}&\frac{-1}{\sqrt{12}}&&&&&&&&&\\ \frac{-1}{\sqrt{12}}&0&\frac{-1}{\sqrt{20}} &&&&&&&& \\ & \frac{-1}{\sqrt{20}} &\frac{-1}{5} &\frac{-1}{5}&&&&&&& \\ &&\frac{-1}{5}&\frac{-1}{5}&\frac{-1}{5}&&&&&& \\ &&&\frac{-1}{5}&\frac{-1}{5}&\frac{-1}{\sqrt{20}}&&&&&\\ &&&&\frac{-1}{\sqrt{20}}&0&\frac{-1}{\sqrt{20}}&&&&\\ &&&&&&\ddots&&&&\\ &&&&&&\frac{-1}{5}&\frac{-1}{5}&\frac{-1}{\sqrt{20}}&&\\ &&&&&&&\frac{-1}{\sqrt{20}}&0&\frac{-1}{\sqrt{20}}&\\ &&&&&&&&\frac{-1}{\sqrt{20}}&\frac{-1}{5}&\frac{-1}{\sqrt{15}}\\ &&&&&&&&&\frac{-1}{\sqrt{15}}&\frac{-1}{3}\\ \end{array} \right)_{(4n)\times(4n)}. \end{eqnarray*}$

Therefore,

$\begin{eqnarray*} \mathcal{L}_{A}& = & \left( \begin{array}{ccccccccccccc} \frac{2}{3}&\frac{-1}{\sqrt{3}}&&&&&&&&&\\ \frac{-1}{\sqrt{3}}&1&\frac{-1}{\sqrt{5}} &&&&&&&& \\ & \frac{-1}{\sqrt{5}} &\frac{4}{5} &\frac{-2}{5}&&&&&&& \\ &&\frac{-2}{5}&\frac{4}{5}&\frac{-2}{5}&&&&&& \\ &&&\frac{-2}{5}&\frac{4}{5}&\frac{-1}{\sqrt{5}}&&&&&\\ &&&&\frac{-1}{\sqrt{5}}&1&\frac{-1}{\sqrt{5}}&&&&\\ &&&&&&\ddots&&&&\\ &&&&&&\frac{-2}{5}&\frac{4}{5}&\frac{-1}{\sqrt{5}}&&\\ &&&&&&&\frac{-1}{\sqrt{5}}&1&\frac{-1}{\sqrt{5}}&\\ &&&&&&&&\frac{-1}{\sqrt{5}}&\frac{4}{5}&\frac{-2}{\sqrt{15}}\\ &&&&&&&&&\frac{-2}{\sqrt{15}}&\frac{2}{3}\\ \end{array} \right)_{(4n)\times(4n)}, \end{eqnarray*}$

and

$\begin{equation*} \mathcal{L}_{S} = diag(\frac{4}{3},\; 1,\; \frac{6}{5},\; \frac{6}{5},\; \frac{6}{5}, \cdot \cdot \cdot ,\; \frac{6}{5},\; 1,\; \frac{6}{5},\; \frac{4}{3}). \end{equation*}$

Assume that the roots of $P_{L_{A}}(x) = 0$ are $0\leq \xi_1 < \xi_2\leq \xi_3 \leq \cdots \leq \alpha_{4n}$ , and $0 \leq \gamma_1 \leq \gamma_2 \leq \gamma_3 \leq \cdots \leq \gamma_{4n}$ are the roots of $P_{L_{S}}(x) = 0$ . By Lemma 2.3, we can get

$\begin{equation*} Kf^{*}(L^{2}_{n}) = 2(19n-4)\Bigg(\sum\limits_{i = 2}^{4n}\frac{1}{\xi_{i}}+\sum\limits_{j = 1}^{4n}\frac{1}{\gamma_{j}}\Bigg). \end{equation*}$

Since $\mathcal{L}_{s}$ is a diagonal matrix. Obviously, its diagonal elements $1, \frac{4}{3}$ and $\frac{6}{5}$ correspond to the eigenvalues of $\mathcal{L}_{s}$ , respectively. Then, it can be clearly obtained as

$\begin{eqnarray} \sum\limits_{i = 1}^{4n}\frac{1}{\gamma_{i}} = \frac{21n-1}{6}. \end{eqnarray}$

(4.1)

Let

$\begin{equation*} P_{\mathcal{L}_{A}}(x) = det(xI-\mathcal{L}_{A}) = x^{4n}+b_{1}x^{4n-1}+\cdots+b_{4n-1}x, b_{4n-1}\ne 0, \end{equation*}$

$\frac{1}{\xi_{2}}, \frac{1}{\xi_{3}}, \cdot\cdot\cdot, \frac{1}{\xi_{4n+1}}$ are the roots of the following equation

$\begin{eqnarray*} b_{4n-1}x^{4n-1}+b_{4n-2}x^{4n-2}+\cdot\cdot\cdot+b_{1}x+1 = 0. \end{eqnarray*}$

Based on the Vieta' s theorem of $P_{L_{A}}(x)$ , we can get

$\begin{equation*} \sum\limits_{i = 2}^{4n}\frac{1}{\xi_{i}} = \frac{(-1)^{4n-2}b_{4n-2}}{(-1)^{4n-1}b_{4n-1}}. \end{equation*}$

Similarly, we can get another two interesting facts.

Theorem 4.1. $(-1)^{4n-1}b_{4n-1} = \frac{25}{9}(38n-8)(\frac{4}{125})^{n}$ .

Proof. Let $s_p = detF_p$ , then we have $s_1 = \frac{2}{3}, s_2 = \frac{1}{3}, s_3 = \frac{2}{15}, s_4 = \frac{4}{75}$ , and

$\begin{eqnarray*} \begin{cases} s_{4p} = \frac{4}{5}s_{4p-1}-\frac{4}{25}s_{4p-2};\\ s_{4p+1} = \frac{4}{5}s_{4p}-\frac{4}{25}s_{4p-1};\\ s_{4p+2} = s_{4p+1}-\frac{1}{5}s_{4p};\\ s_{4p+3} = \frac{4}{5}s_{4p+2}-\frac{1}{5}s_{4p+1}.\\ \end{cases} \end{eqnarray*}$

After further simplification, the transformation form of the above formula is obtained.

$\begin{eqnarray*} \begin{cases} s_{4p} = \frac{5}{3}\cdot(\frac{4}{125})^{p}, 1\leq p\leq n;\\ s_{4p+1} = \frac{2}{3}\cdot(\frac{4}{125})^{i}, 0\leq p\leq n-1;\\ s_{4p+2} = \frac{1}{3}\cdot(\frac{4}{125})^{i}, 0\leq p\leq n-1;\\ s_{4p+3} = \frac{1}{15}\cdot(\frac{4}{125})^{i}, 0\leq p\leq n-1. \end{cases} \end{eqnarray*}$

Similarly, we have $t_1 = \frac{2}{3}, t_2 = \frac{4}{15}, t_3 = \frac{2}{15}, t_4 = \frac{4}{75}$ , and

$\begin{eqnarray*} \begin{cases} t_{4p} = \frac{2}{5}t_{4p-1}-\frac{2}{5}t_{4p-2};\\ t_{4p+1} = \frac{4}{5}t_{4p}-\frac{4}{25}t_{4p-1};\\ t_{4p+2} = \frac{4}{5}t_{4p+1}-\frac{4}{25}t_{4p};\\ t_{4p+3} = t_{4p+2}-\frac{1}{5}t_{4p+1}.\\ \end{cases} \end{eqnarray*}$

Therefore, the transformation form of the above formula is obtained.

$\begin{eqnarray*} \begin{cases} t_{4p-4} = \frac{5}{3}\cdot(\frac{4}{125})^{p}, 1\leq p\leq n;\\ t_{4p-3} = \frac{2}{3}\cdot(\frac{4}{125})^{p}, 0\leq p\leq n-1;\\ t_{4p-2} = \frac{4}{15}\cdot(\frac{4}{125})^{p}, 0\leq p\leq n-1;\\ t_{4p-1} = \frac{2}{15}\cdot(\frac{4}{125})^{p}, 0\leq p\leq n-1. \end{cases} \end{eqnarray*}$

Since the $(-1)^{4n-1}b_{4n-1}$ is the total of all the principal minors of order $4n-1$ of $L_{A}$ , we have

$\begin{eqnarray*} (-1)^{4n-1}b_{4n-1}& = &\sum^{4n}_{i = 2}detNL_{A}[i]+s_{4n}+t_{4n}\\ & = &\frac{1}{45}(38n-8)(\frac{4}{125})^{n}. \end{eqnarray*}$

The proof of Theorem 4.1 completed. □

Theorem 4.2. $(-1)^{4n-2}b_{4n-2} = \frac{1}{3240}(14520n^3+4599n^2-1496n+3)(\frac{4}{125})^{n}$ .

Proof. We observe that the sum of all the principal minors of order $4n$ in $\mathcal{L}_{A}$ is $(-1)^{4n-2}b_{4n-2}$ , then

$\begin{eqnarray} (-1)^{4n-2}b_{4n-2} = \sum\limits_{1 \leq s < t \leq 4n}det \mathcal{L}_{A}[s,t]\cdot f_{s-1} \cdot f_{4n-t}^{'}. \end{eqnarray}$

(4.2)

By Eq (4.8), we know that the result of $det\mathcal{L}_{A}[s, t]$ will change with the values of $s$ and $t$ . Then we can get the following twenty cases:

Case 1. $i = 4s$ , $j = 4t$ , $1\leq{s} < t\leq{n}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{2}{\sqrt{5}} & 1 & -\frac{1}{5} & & & && \\ & -\frac{1}{5} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & \frac{-2}{5} &\frac{4}{5}& \frac{-2}{5}&&&\\ & && & \frac{-2}{5} & -\frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{5} & 1&-\frac{1}{\sqrt{5}}&\\ &&&&&&\frac{-2}{5}&-\frac{4}{5}\\ \end{array} \right|_{(4t-4s-1)}\\ & = &10(t-s)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 2. $i = 4s$ , $j = 4t+1$ , $1\leq{s}\leq t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & 1 & -\frac{1}{5} & & & && \\ & -\frac{1}{5} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & \frac{-2}{5} &\frac{4}{5}& \frac{-1}{5}&&&\\ & && & \frac{-1}{5} & 1&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & \frac{4}{5}&-\frac{2}{\sqrt{5}}&\\ &&&&&&\frac{-2}{5}&-\frac{4}{5}\\ \end{array} \right|_{(4t-4s)}\\ & = &(4t-4s+1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 3. $i = 4s$ , $j = 4t+2$ , $1\leq{s}\leq t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & 1 & -\frac{1}{5} & & & && \\ & -\frac{1}{5} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &1&-\frac{1}{\sqrt{5}} &&&\\ & && & -\frac{1}{\sqrt{5}} & \frac{4}{5}&\frac{-2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}&-\frac{2}{5}&\\ &&&&&&\frac{-2}{5}&-\frac{4}{5}\\ \end{array} \right|_{(4t-4s+1)}\\ & = &\frac{4}{5}(2t-2s+1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 4. $i = 4s$ , $j = 4t+3$ , $1\leq{s}\leq t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & 1 & -\frac{1}{5} & & & && \\ & -\frac{1}{5} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&\frac{-2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&\\ &&&&&&-\frac{1}{\sqrt{5}}&1\\ \end{array} \right|_{(4t-4s+2)}\\ & = &\frac{1}{5}(4t-4s+3)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 5. $i\equiv 0$ , $j = 4n$ , $1\leq{s}\leq t$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{2}{\sqrt{5}} & 1 & -\frac{1}{5} & & & && \\ & -\frac{1}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&\frac{-1}{\sqrt{5}}&&\\ & & & & &\frac{-1}{\sqrt{5}} & 1&-\frac{1}{\sqrt{5}}&\\ &&&&&&-\frac{2}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4n-4s-1)}\\ & = &10(n-s)\Big(\frac{4}{125}\Big)^{n-s}. \end{eqnarray*}$

Case 6. $i = 4s+1$ , $j = 4t$ , $1\leq{s} < t\leq{n}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} 1 & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & \frac{4}{5} & -\frac{2}{\sqrt{5}} & & & && \\ & -\frac{2}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1&-\frac{1}{\sqrt{5}}&\\ &&&&&&-\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-2)}\\ & = &\frac{25}{4}(4t-4s-1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 7. $i = 4s+1$ , $j = 4t+1$ , $1\leq{s} < t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} 1 & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & \frac{4}{5} & -\frac{2}{\sqrt{5}} & & & && \\ & -\frac{2}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{1}{\sqrt{5}} &&&\\ & && & -\frac{1}{\sqrt{5}} & 1&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & \frac{4}{5}& -\frac{2}{5}&\\ &&&&&& -\frac{2}{5}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-1)}\\ & = &10(t-s)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 8. $i = 4s+1$ , $j = 4t+2$ , $1\leq{s} < t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} 1 & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & \frac{4}{5} & -\frac{2}{\sqrt{5}} & & & && \\ & -\frac{2}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &1&-\frac{1}{\sqrt{5}} &&&\\ & && & -\frac{1}{\sqrt{5}} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{2}{5}&\\ &&&&&& -\frac{2}{5}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s)}\\ & = &(4t-4s+1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 9. $i = 4s+1$ , $j = 4t+3$ , $0\leq{s} \leq t\leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} 1 & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & \frac{4}{5} & -\frac{2}{\sqrt{5}} & & & && \\ & -\frac{2}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&1\\ \end{array} \right|_{(4t-4s+1)}\\ & = &(2t-2s+1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 10. $i\equiv 1$ , $j = 4n+1$ , $0\leq{s} \leq{n}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} 1 & -\frac{1}{\sqrt{5}} & & & & && \\ -\frac{1}{\sqrt{5}} & \frac{4}{5} & -\frac{2}{\sqrt{5}} & & & && \\ & -\frac{2}{\sqrt{5}} &\frac{4}{5} & -\frac{2}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4n-4s-2)}\\ & = &\frac{25}{4}(4n-4s-1)\Big(\frac{4}{125}\Big)^{n-s}. \end{eqnarray*}$

Case 11. $i = 4s+2$ , $j = 4t$ , $0\leq{s} < t \leq{n}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{2}{5} & & & && \\ & -\frac{2}{5} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5} &&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-3)}\\ & = &25(2t-2s-1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 12. $i = 4s+2$ , $j = 4t+1$ , $0\leq{s} < t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{2}{5} & & & && \\ & -\frac{2}{5} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &1&-\frac{1}{\sqrt{5}}&&&\\ & && & -\frac{1}{\sqrt{5}} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{2}{5}&\\ &&&&&& -\frac{2}{5}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-2)}\\ & = &5(4t-4s-1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 13. $i = 4s+2$ , $j = 4t+2$ , $0\leq{s} < t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{2}{5} & & & && \\ & -\frac{2}{5} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &1&-\frac{1}{\sqrt{5}}&&&\\ & && & -\frac{1}{\sqrt{5}} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{2}{5}&\\ &&&&&& -\frac{2}{5}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-1)}\\ & = &8(t-s)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 14. $i = 4s+2$ , $j = 4t+3$ , $0\leq{s}\leq t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{2}{5} & & & && \\ & -\frac{2}{5} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &\frac{4}{5}&-\frac{2}{5}&&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&1\\ \end{array} \right|_{(4t-4s)}\\ & = &(4t-4s+1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 15. $i\equiv 2$ , $j = 4t+3$ , $0\leq{s} \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{2}{5} & & & && \\ & -\frac{2}{5} &\frac{4}{5} & -\frac{2}{5} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5}&&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4n-4s-3)}\\ & = &25(2n-2s-1)\Big(\frac{4}{125}\Big)^{n-s}. \end{eqnarray*}$

Case 16. $i = 4s+3$ , $j = 4t$ , $0\leq{s} < t \leq{n}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{1}{\sqrt{5}} & & & && \\ & -\frac{1}{\sqrt{5}} &1 & -\frac{1}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5}&&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-4)}\\ & = &\frac{125}{4}(4t-4s-3)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 17. $i = 4s+3$ , $j = 4t+1$ , $0\leq{s} < t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{1}{\sqrt{5}} & & & && \\ & -\frac{1}{\sqrt{5}} &1 & -\frac{1}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{1}{\sqrt{5}}&&&\\ & && & -\frac{1}{\sqrt{5}} & 1&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & \frac{4}{5}& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-3)}\\ & = &25(2t-2s-1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 18. $i = 4s+3$ , $j = 4t+2$ , $0\leq{s} < t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{1}{\sqrt{5}} & & & && \\ & -\frac{1}{\sqrt{5}} &1 & -\frac{1}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &1&-\frac{1}{\sqrt{5}}&&&\\ & && & -\frac{1}{\sqrt{5}} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}&-\frac{2}{5}&\\ &&&&&& -\frac{2}{5}&\frac{4}{5}\\ \end{array} \right|_{(4t-4s-3)}\\ & = &\frac{25}{3}(4t-4s-1)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 19. $i = 4s+3$ , $j = 4t+3$ , $0\leq{s} < t \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{1}{\sqrt{5}} & & & && \\ & -\frac{1}{\sqrt{5}} &1 & -\frac{1}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{1}{\sqrt{5}} &\frac{4}{5}&-\frac{2}{5}&&&\\ & && & -\frac{2}{5} & \frac{4}{5}&-\frac{2}{5}&&\\ & & & & &-\frac{2}{5} & \frac{4}{5}& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&1\\ \end{array} \right|_{(4t-4s-1)}\\ & = &10(t-s)\Big(\frac{4}{125}\Big)^{t-s}. \end{eqnarray*}$

Case 20. $i\equiv 3$ , $j = 4t$ , $0\leq{s} \leq{n-1}$ ,

$\begin{eqnarray*} det\; \varphi& = & \left| \begin{array}{ccccccccc} \frac{4}{5} & -\frac{2}{5} & & & & && \\ -\frac{2}{5} & \frac{4}{5} &-\frac{1}{\sqrt{5}} & & & && \\ & -\frac{1}{\sqrt{5}} &1 & -\frac{1}{\sqrt{5}} &&&&\\ &&&\ddots&&&&\\ & & & -\frac{2}{5} &\frac{4}{5}&-\frac{2}{5}&&&\\ & && & -\frac{2}{5} & 1&-\frac{1}{\sqrt{5}}&&\\ & & & & &-\frac{1}{\sqrt{5}} & 1& -\frac{1}{\sqrt{5}}&\\ &&&&&& -\frac{1}{\sqrt{5}}&\frac{4}{5}\\ \end{array} \right|_{(4n-4s-4)}\\ & = &\frac{125}{4}(4n-4s-3)\Big(\frac{4}{125}\Big)^{n-s}. \end{eqnarray*}$

Therefore, we can get

$\begin{eqnarray*} (-1)^{4n-2}b_{4n-2} = E_1+E_2+E_3+E_4, \end{eqnarray*}$

where

$\begin{eqnarray*} E_1 = \frac{1}{18}(227n^3+347n^2-574n+4)\Big(\frac{4}{125}\Big)^{n-1}. \end{eqnarray*}$

$\begin{eqnarray*} E_2 = \frac{1}{72}(908n^3+3431n^2+523n)\Big(\frac{4}{125}\Big)^{n}. \end{eqnarray*}$

$\begin{eqnarray*} E_3 = \frac{1}{45}(454n^3+1375n^2-1079n)\Big(\frac{4}{125}\Big)^{n}. \end{eqnarray*}$

$\begin{eqnarray*} E_4 = \frac{1}{81}(92n^3+561n^2-611n)\Big(\frac{4}{125}\Big)^{n-1}. \end{eqnarray*}$

Hence

$\begin{eqnarray*} (-1)^{4n-2}b_{4n-2} = E_1+E_2+E_3+E_4 = \frac{1}{3240}(14520n^3+4599n^2-1496n+4)\Big(\frac{4}{125}\Big)^{n}. \end{eqnarray*}$

The proof of Theorem 4.2 completed. □

Let $0 \leq \xi_1 \leq \xi_2 \leq \xi_3 \leq \cdots \leq \xi_{3n+2}$ are the eigenvalues of $\mathscr{L}_{A}$ . We can get the following exact equation:

$\begin{equation*} \sum\limits_{i = 2}^{4n}\frac{1}{\xi_{i}} = \frac{(-1)^{4n-2}b_{4n-2}}{(-1)^{4n-1}b_{4n-1}} = \frac{1}{72}\Big(\frac{14520n^3+4599n^2-1496n+8}{38n-8}\Big). \end{equation*}$

Theorem 4.3. Set $L_{n}^{6, 4, 4}$ be the derivative [n]pheylenes, and the expression of the multiplicative degree-Kirchhoff index is

$\begin{equation*} Kf^{*}(L_{n}) = \Big(\frac{29040n^3+8996n^2-3198n+8}{144}\Big). \end{equation*}$

Proof. Together with Eq.(4.7) and Theorems 4.1 and 4.2, one can get

$\begin{eqnarray*} Kf^{*}(L_{n})& = &2(19n-4)\Big(\sum\limits_{i = 2}^{4n}\frac{1}{\xi_{i}}+\sum\limits_{i = 1}^{4n}\frac{1}{\gamma_{i}}\Big)\\ & = &2(19n-4)[\frac{1}{72}\Big(\frac{14520n^3+4599n^2-1496n+8}{38n-8}\Big)+\frac{21n-1}{6}]\\ & = &\Big(\frac{29040n^3+8996n^2-3198n+8}{144}\Big). \end{eqnarray*}$

The result as desired. □

The multiplicative degree-Kirchhoff indices of $L_n$ from $L_1$ to $L_{12}$ , see Table 2.

Table 2. The multiplicative degree-Kirchhoff indices of

$L_{1}, \; L_{2}, ..., \; L_{12}.$ .

$\mathscr{G}$	$Kf^{*}(\mathscr{G})$	$\mathscr{G}$	$Kf^{*}(\mathscr{G})$	$\mathscr{G}$	$Kf^{*}(\mathscr{G})$
$L_1$	$241.98$	$L_5$	$26659.15$	$L_9$	$151875.4$
$L_2$	$1818.86$	$L_6$	$45675.81$	$L_{10}$	$207691.9$
$L_3$	$5940.68$	$L_7$	$72077.4$	$L_{11}$	$275733.2$
$L_4$	$13817.44$	$L_8$	$107073.9$	$L_{12}$	$357209.6$

| Show Table

DownLoad: CSV

Then we want to calculate the Gutman index of $L_n$ .

Theorem 4.4. Suppose that $L_{n}^{6, 4, 4}$ is the dicyclobutadieno derivative of [n]phenylenes and the graph $L_{n}$ is obtained from the transformation of the graph $L_{n}^{6, 4, 4}$ , then

$\begin{eqnarray*} \lim\limits_{n \to \infty}\frac{Kf^{*}(L_n)}{Gut(L_n)} = \frac{1}{4}. \end{eqnarray*}$

Proof. Consider $d_{ij}$ for all vertices. We divide the vertices of $L_n$ into the following four categories.

Case 1. Vertex $4i-2(i = 1, 2, \cdots, n)$ of $L_{n}$ :

$\begin{eqnarray*} f_{4i-2} = \frac{10}{3}n(56n^2-24n+37). \end{eqnarray*}$

Case 2. Vertex $4i-1(i = 1, 2, \cdots, n)$ of $L_{n}$ :

$\begin{eqnarray*} f_{4i-1} = \frac{10}{3}n(152n^2-48n+29). \end{eqnarray*}$

Case 3. Vertex $4i(i = 1, 2, \cdots, n)$ of $L_{n}$ :

$\begin{eqnarray*} f_{4i} = \frac{10}{3}n(140n^2-48n+43). \end{eqnarray*}$

Case 4. Vertex $4i-3(i = 1, 2, \cdots, n)$ of $L_{n}$ :

$\begin{eqnarray*} f_{4i-3} = \frac{10}{3}n(136n^2-6n+71). \end{eqnarray*}$

According to Eq.(1.3), the Gutman index of $L_n$ is

$\begin{eqnarray*} Gut(L_{n})& = &\frac{f_{4i}+f_{4i-1}+f_{4i-2}+f_{4i-3}}{2}\\ & = &\frac{10n}{3}(242n^2-63n+61). \end{eqnarray*}$

Therefore, combining with $Kf^{*}(L_n)$ and $Gut(L_n)$ , we have

$\begin{eqnarray*} \lim\limits_{n \to \infty}\frac{Kf^{*}(L_n)}{Gut(L_n)} = \frac{1}{4}. \end{eqnarray*}$

The result as desired.□

Finally, we want to know the complexity of $L_n$ .

Theorem 4.5. For the graph $L_n$ , we have

$\begin{eqnarray*} \tau(L_n) = 2^{3n+2}\cdot 3^{3n-2}. \end{eqnarray*}$

Proof. Based on Lemma 2.4, we can get

$\begin{eqnarray*} \prod\limits_{i = 1}^{8n}d_{i}\prod\limits_{i = 2}^{4n}\alpha_{i}\prod\limits_{j = 1}^{4n}\beta_{i} = 2(19n-4)\cdot \tau(L_n). \end{eqnarray*}$

Note that

$\begin{eqnarray*} \prod\limits_{i = 1}^{8n}d_{i} = 3^4\cdot 4^{2n}\cdot 5^{6n-4}. \end{eqnarray*}$

$\begin{eqnarray*} \prod\limits_{i = 2}^{4n}\alpha_{i} = \frac{25}{9}\cdot (38n-8)\cdot (\frac{4}{125})^{n}. \end{eqnarray*}$

$\begin{eqnarray*} \prod\limits_{j = 1}^{4n}\beta_{i} = (\frac{4}{3})^2\cdot (\frac{6}{5})^{3n-2}. \end{eqnarray*}$

Hence,

$\begin{eqnarray*} \tau(L_n) = 2^{3n+2}\cdot 3^{3n-2}. \end{eqnarray*}$

The proof is over. □

Thus, we can get the complexity of $L_n$ from $W_1$ to $W_{8}$ which are listed in Table 3.

Table 3. The complexity of

$W_1, W_2 \cdots W_8$ .

$\mathscr{G}$	$\tau{(\mathscr{G})}$	$\mathscr{G}$	$\tau{(\mathscr{G})}$
$W_1$	$96$	$W_5$	$208971104256$
$W_2$	$20736$	$W_6$	$45137758519296$
$W_3$	$4478976$	$W_7$	$9749755840167936$
$W_4$	$967458816$	$W_8$	$2105947261476274176$

| Show Table

DownLoad: CSV

5. Conclusions

In this paper, the linear chain network with $n$ hexagons and $2n-1$ squares is considered. We have devoted ourselves to calculating the (multiplicative degree) Kirchhoff index, Wiener index, Gutman index, and complexity. In the meantime, we deduced that the ratio of the (multiplicative degree) Kirchhoff index to the (Gutman) Wiener index is nearly a quarter when $n$ tends to infinity. These rules also apply to some other graphs.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by the Anhui Provincial Natural Science Research Major Project (No. 2022AH040317) and the Anhui Provincial 2023 Action Project for Cultivating Young and Middle Aged Teachers in Universities (No. DTR2023095).

Conflict of interest

No potential conflicts of interest were reported by the authors.

References

[1]	Q. Ai, J. Huang, S. Du, K. Yang, H. Wang, Comprehensive evaluation of very thin asphalt overlays with different aggregate gradations and asphalt materials based on AHP and TOPSIS, Buildings, 12 (2022), 1149. https://doi.org/10.3390/buildings12081149 doi: 10.3390/buildings12081149
[2]	W. Li, M. Xiao, A. Garg, L. Gao, A new approach to solve uncertain multidisciplinary design optimization based on conditional value at risk, IEEE T. Autom. Sci. Eng., 18 (2021), 356–368. https://doi.org/10.1109/TASE.2020.2999380 doi: 10.1109/TASE.2020.2999380
[3]	D. Meng, S. Yang, C. He, H. Wang, Z. Lv, Y. Guo, et al., Multidisciplinary design optimization of engineering systems under uncertainty: a review, Int. J. Struct. Integr., 13 (2022), 565–593. https://doi.org/10.1108/IJSI-05-2022-0076 doi: 10.1108/IJSI-05-2022-0076
[4]	Q. Zhu, H. Wang, Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function, Automatica, 87 (2018), 166–175. https://doi.org/10.1016/j.automatica.2017.10.004 doi: 10.1016/j.automatica.2017.10.004
[5]	D. Meng, Z. Lv, S. Yang, H. Wang, T. Xie, Z. Wang, A time-varying mechanical structure reliability analysis method based on performance degradation, Structures, 34 (2021), 3247–3256. https://doi.org/10.1016/j.istruc.2021.09.085 doi: 10.1016/j.istruc.2021.09.085
[6]	X. Gao, X. Su, H. Qian, X. Pan, Dependence assessment in human reliability analysis under uncertain and dynamic situations, Nucl. Eng. Technol., 54 (2022), 948–958. https://doi.org/10.1016/j.net.2021.09.045 doi: 10.1016/j.net.2021.09.045
[7]	B. Wang, Q. Zhu, S. Li, Stabilization of discrete-time hidden semi-Markov jump linear systems with partly unknown emission probability matrix, IEEE T. Automat. Contr., 69 (2023), 1952–1959. https://doi.org/10.1109/TAC.2023.3272190 doi: 10.1109/TAC.2023.3272190
[8]	K. Liao, Y. Wu, F. Miao, L. Li, Y. Xue, Time-varying reliability analysis of Majiagou landslide based on weakening of hydro-fluctuation belt under wetting-drying cycles, Landslides, 18 (2021), 267–280. https://doi.org/10.1007/s10346-020-01496-2 doi: 10.1007/s10346-020-01496-2
[9]	K. Gao, G. Liu, Novel nonlinear time-varying fatigue reliability analysis based on the probability density evolution method, Int. J. Fatigue, 149 (2021), 106257. https://doi.org/10.1016/j.ijfatigue.2021.106257 doi: 10.1016/j.ijfatigue.2021.106257
[10]	Y. Zhao, Q. Zhu, Stability of highly nonlinear neutral stochastic delay systems with non-random switching signals, Syst. Control Lett., 165 (2022), 105261. https://doi.org/10.1016/j.sysconle.2022.105261 doi: 10.1016/j.sysconle.2022.105261
[11]	S. Yang, Z. He, J. Chai, D. Meng, W. Macek, R. Branco, et al., A novel hybrid adaptive framework for support vector machine-based reliability analysis: A comparative study, Structures, 58 (2023), 105665. https://doi.org/10.1016/j.istruc.2023.105665 doi: 10.1016/j.istruc.2023.105665
[12]	S. Yang, D. Meng, H. Wang, Z. Chen, B. Xu, A comparative study for adaptive surrogate-model-based reliability evaluation method of automobile components, Int. J. Struct. Integr., 14 (2023), 498–519. https://doi.org/10.1108/IJSI-03-2023-0020 doi: 10.1108/IJSI-03-2023-0020
[13]	C. Luo, S. P. Zhu, B. Keshtegar, X. Niu, O. Taylan, An enhanced uniform simulation approach coupled with SVR for efficient structural reliability analysis, Reliab. Eng. Syst. Safe., 237 (2023), 109377. https://doi.org/10.1016/j.ress.2023.109377 doi: 10.1016/j.ress.2023.109377
[14]	D. Zhang, P. Zhou, C. Jiang, M. Yang, X. Han, Q. Li, A stochastic process discretization method combing active learning Kriging model for efficient time-variant reliability analysis, Comput. Method. Appl. M., 384 (2021), 113990. https://doi.org/10.1016/j.cma.2021.113990 doi: 10.1016/j.cma.2021.113990
[15]	Y. Zhao, D. Zhang, M. Yang, F. Wang, X. Han, On efficient time-dependent reliability analysis method through most probable point-oriented Kriging model combined with importance sampling, Struct. Multidisc. Optim., 67 (2024), 6. https://doi.org/10.1007/s00158-023-03721-7 doi: 10.1007/s00158-023-03721-7
[16]	F. Xiao, Generalized quantum evidence theory, Appl. Intell., 53 (2023), 14329–14344. https://doi.org/10.1007/s10489-022-04181-0 doi: 10.1007/s10489-022-04181-0
[17]	C. Luo, S. P. Zhu, B. Keshtegar, X. Niu, O. Taylan, An enhanced uniform simulation approach coupled with SVR for efficient structural reliability analysis, Reliab. Eng. Syst. Safe., 237 (2023), 109377. https://doi.org/10.1016/j.ress.2023.109377 doi: 10.1016/j.ress.2023.109377
[18]	Z. Meng, Q. Qian, M. Xu, B. Yu, A. R. Yıldız, S. Mirjalili, PINN-FORM: A new physics-informed neural network for reliability analysis with partial differential equation, Comput. Method. Appl. M., 414 (2023), 116172. https://doi.org/10.1016/j.cma.2023.116172 doi: 10.1016/j.cma.2023.116172
[19]	F. Kong, H. Ni, Q. Zhu, C. Hu, T. Huang, Fixed-time and predefined-time synchronization of discontinuous neutral-type competitive networks via non-chattering adaptive control strategy, IEEE T. Netw. Sci. Eng., 10 (2023), 3644–3657. https://doi.org/10.1109/TNSE.2023.3271109 doi: 10.1109/TNSE.2023.3271109
[20]	Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Le´vy processes, IEEE T. Automat. Contr., 2024. https://doi.org/10.1109/TAC.2024.3448128
[21]	D. Meng, S. Yang, A. M. De. Jesus, S. P. Zhu, A novel Kriging-model-assisted reliability-based multidisciplinary design optimization strategy and its application in the offshore wind turbine tower, Renew. Energ., 203 (2023), 407–420. https://doi.org/10.1016/j.renene.2022.12.062 doi: 10.1016/j.renene.2022.12.062
[22]	C. Luo, S. P. Zhu, B. Keshtegar, W. Macek, R. Branco, D. Meng, Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy, Comput. Method. Appl. M., 423 (2024), 116863. https://doi.org/10.1016/j.cma.2024.116863 doi: 10.1016/j.cma.2024.116863
[23]	Y. Xue, Y. Deng, Extending set measures to orthopair fuzzy sets, Int. J. Uncertain. Fuzz., 30 (2022), 63–91. https://doi.org/10.1142/S0218488522500040 doi: 10.1142/S0218488522500040
[24]	W. Peng, Y. F. Li, Y. J. Yang, J. Mi, H. Z. Huang, Bayesian degradation analysis with inverse Gaussian process models under time-varying degradation rates, IEEE T. Reliab., 66 (2017), 84–96. https://doi.org/10.1109/TR.2016.2635149 doi: 10.1109/TR.2016.2635149
[25]	H. V. Dang, R. Trestian, T. Bui-Tien, H. X. Nguyen, Probabilistic method for time-varying reliability analysis of structure via variational Bayesian neural network, Structures, 34 (2021), 3703–3715. https://doi.org/10.1016/j.istruc.2021.09.069 doi: 10.1016/j.istruc.2021.09.069
[26]	C. Jiang, D. Wang, H. Qiu, L. Gao, L. Chen, Z. Yang, An active failure-pursuing Kriging modeling method for time-dependent reliability analysis, Mech. Syst. Signal Pr., 129 (2019), 112–129. https://doi.org/10.1016/j.ymssp.2019.04.034 doi: 10.1016/j.ymssp.2019.04.034
[27]	W. Han, X. Zhang, X. Huang, H. Li, A time-dependent reliability estimation method based on Gaussian process regression, In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2018. https://doi.org/10.1115/DETC2018-86294
[28]	S. Yu, Z. Wang, Y. Li, Time and space-variant system reliability analysis through adaptive Kriging and weighted sampling, Mech. Syst. Signal Pr., 166 (2022), 108443. https://doi.org/10.1016/j.ymssp.2021.108443 doi: 10.1016/j.ymssp.2021.108443
[29]	D. Meng, H. Yang, S. Yang, Y. Zhang, A. M. De. Jesus, J. Correia, et al., Kriging-assisted hybrid reliability design and optimization of offshore wind turbine support structure based on a portfolio allocation strategy, Ocean Eng., 295 (2024), 116842. https://doi.org/10.1016/j.oceaneng.2024.116842 doi: 10.1016/j.oceaneng.2024.116842
[30]	Y. Pang, Y. Wang, X. Lai, S. Zhang, P. Liang, X. Song, Enhanced Kriging leave-one-out cross-validation in improving model estimation and optimization, Comput. Method. Appl. M., 414 (2023), 116194. https://doi.org/10.1016/j.cma.2023.116194 doi: 10.1016/j.cma.2023.116194
[31]	J. Xue, B. Shen, Dung beetle optimizer: A new meta-heuristic algorithm for global optimization, J. Supercomput., 79 (2023), 7305–7336. https://doi.org/10.1007/s11227-022-04959-6 doi: 10.1007/s11227-022-04959-6
[32]	D. Zhang, G. Li, Z. Bie, K. Fan, An analytical method for reliability evaluation of power distribution system with time-varying failure rates, Reliab. Eng. Syst. Safe., 250 (2024), 110290. https://doi.org/10.1016/j.ress.2024.110290 doi: 10.1016/j.ress.2024.110290
[33]	D. Meng, S. Yang, A. M. De. Jesus, T. Fazeres-Ferradosa, S. P. Zhu, A novel hybrid adaptive Kriging and water cycle algorithm for reliability-based design and optimization strategy: Application in offshore wind turbine monopile, Comput. Method. Appl. M., 412 (2023), 116083. https://doi.org/10.1016/j.cma.2023.116083 doi: 10.1016/j.cma.2023.116083
[34]	Z. Yuan, J. Li, H. Yang, B. Zhang, A hybrid whale optimization and particle swarm optimization algorithm, In: 2021 IEEE International Conference on Progress in Informatics and Computing (PIC), 2021,260–264. https://doi.org/10.1109/PIC53636.2021.9687017
[35]	Bilal, M. Pant, H. Zaheer, L. Garcia-Hernandez, A. Abraham, Differential evolution: A review of more than two decades of research, Eng. Appl. Artif. Intel., 90 (2020), 103479. https://doi.org/10.1016/j.engappai.2020.103479 doi: 10.1016/j.engappai.2020.103479
[36]	Y. del. Valle, G. K. Venayagamoorthy, S. Mohagheghi, J. C. Hernandez, R. G. Harley, Particle swarm optimization: basic concepts, variants and applications in power systems, IEEE T. Evolut. Comput., 12 (2008), 171–195. https://doi.org/10.1109/TEVC.2007.896686 doi: 10.1109/TEVC.2007.896686
[37]	S. Yang, H. Wang, Y. Xu, Y. Guo, L. Pan, J, Zhang, et al., A coupled simulated annealing and particle swarm optimization reliability-based design optimization strategy under hybrid uncertainties, Mathematics, 11 (2023), 4790. https://doi.org/10.3390/math11234790 doi: 10.3390/math11234790
[38]	N. Hansen, D. V. Arnold, A. Auger, Evolution strategies, In: Springer Handbook of computational intelligence, 2015,871–898. https://doi.org/10.1007/978-3-662-43505-2_44
[39]	B. Yang, J. Wang, X. Zhang, T. Yu, W. Yao, H. Shu, et al., Comprehensive overview of meta-heuristic algorithm applications on PV cell parameter identification, Energ. Convers. Manage., 208 (2020), 112595. https://doi.org/10.1016/j.enconman.2020.112595 doi: 10.1016/j.enconman.2020.112595
[40]	G. Khanna, S. K. Chaturvedi, S. Soh, Time varying communication networks: Modelling, reliability evaluation and optimization, In: Advances in reliability analysis and its applications, 2020. https://doi.org/10.1007/978-3-030-31375-3_1
[41]	J. Drugowitsch, R. Moreno-Bote, A. Pouget, Optimal decision-making with time-varying evidence reliability, Adv. Neural Inform. Process. Syst., 27 (2014).
[42]	J. H. Holland, Genetic algorithms, Sci. Am., 267 (1992), 66–73.
[43]	S. Mirjalili, A. Lewis, The whale optimization algorithm, Adv. Eng. Softw., 95 (2016), 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008 doi: 10.1016/j.advengsoft.2016.01.008
[44]	S. P. Zhu, B. Keshtegar, M. E. A. B. Seghier, E. Zio, O. Taylan, Hybrid and enhanced PSO: Novel first order reliability method-based hybrid intelligent approaches, Comput. Method. Appl. M., 393 (2022), 114730. https://doi.org/10.1016/j.cma.2022.114730 doi: 10.1016/j.cma.2022.114730
[45]	R. Poli, J. Kennedy, T. Blackwell, Particle swarm optimization: An overview, Swarm Intell., 1 (2007), 33–57. https://doi.org/10.1007/s11721-007-0002-0 doi: 10.1007/s11721-007-0002-0
[46]	H. Gao, S. Kwong, J. Yang, J. Cao, Particle swarm optimization based on intermediate disturbance strategy algorithm and its application in multi-threshold image segmentation, Inform. Sciences, 250 (2013), 82–112. https://doi.org/10.1016/j.ins.2013.07.005 doi: 10.1016/j.ins.2013.07.005
[47]	M. Pant, H. Zaheer, L. Garcia-Hernandez, A. Abraham, Differential evolution: A review of more than two decades of research, Eng. Appl. Artif. Intel., 90 (2020), 103479. https://doi.org/10.1016/j.engappai.2020.103479 doi: 10.1016/j.engappai.2020.103479
[48]	S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey wolf optimizer, Adv. Eng. Softw., 69 (2014), 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007 doi: 10.1016/j.advengsoft.2013.12.007
[49]	Z. Dong, Z. Sheng, Y. Zhao, P. Zhi, Robust optimization design method for structural reliability based on active-learning MPA-BP neural network, Int. J. Struct. Integr., 14 (2023), 248–266. https://doi.org/10.1108/IJSI-10-2022-0129 doi: 10.1108/IJSI-10-2022-0129
[50]	D. Meng, H. Yang, S. Yang, Y. Zhang, A. M. De. Jesus, J. Correia, et al., Kriging-assisted hybrid reliability design and optimization of offshore wind turbine support structure based on a portfolio allocation strategy, Ocean Eng., 295 (2024), 116842. https://doi.org/10.1016/j.oceaneng.2024.116842 doi: 10.1016/j.oceaneng.2024.116842
[51]	Z. Zhu, Q. Zhu, Adaptive event-triggered fuzzy control for stochastic highly nonlinear systems with time delay and nontriangular structure interconnections, IEEE T. Fuzzy Syst., 32 (2023), 27–37. https://doi.org/10.1109/TFUZZ.2023.3287869 doi: 10.1109/TFUZZ.2023.3287869
[52]	N. Hansen, D. V. Arnold, A. Auger, Evolution strategies, In: Springer Handbook of computational intelligence, 2015. https://doi.org/10.1007/978-3-662-43505-2_44
[53]	F. Kong, H. Ni, Q. Zhu, C. Hu, T. Huang, Fixed-time and predefined-time synchronization of discontinuous neutral-type competitive networks via non-chattering adaptive control strategy, IEEE T. Netw. Sci. Eng., 10 (2023), 3644–3657. https://doi.org/10.1109/TNSE.2023.3271109 doi: 10.1109/TNSE.2023.3271109
[54]	Z. Wang, P. Wang, A new approach for reliability analysis with time-variant performance characteristics, Reliab. Eng. Syst. Safe., 115 (2013), 70–81. https://doi.org/10.1016/j.ress.2013.02.017 doi: 10.1016/j.ress.2013.02.017
[55]	S. Yang, C. Guo, D. Meng, Y. Guo, Y. Guo, L. Pan, et al., MECSBO: Multi‐strategy enhanced circulatory system based optimisation algorithm for global optimisation and reliability-based design optimisation problems, IET Coll. Intell. Manuf., 6 (2024), e12097. https://doi.org/10.1049/cim2.12097 doi: 10.1049/cim2.12097
[56]	S. Yang, H. Wang, Y. Xu, Y. Guo, L. Pan, J. Zhang, et al., A coupled simulated annealing and particle swarm optimization reliability-based design optimization strategy under hybrid uncertainties, Mathematics, 11 (2023), 4790. https://doi.org/10.3390/math11234790 doi: 10.3390/math11234790
[57]	C. Gong, D. M. Frangopol, An efficient time-dependent reliability method, Struct. Saf., 81 (2019), 101864. https://doi.org/10.1016/j.strusafe.2019.05.001 doi: 10.1016/j.strusafe.2019.05.001
[58]	C. Jiang, X. P. Huang, X. Han, D. Q. Zhang, A time-variant reliability analysis method based on stochastic process discretization, J. Mech. Des., 136 (2014), 091009. https://doi.org/10.1115/1.4027865 doi: 10.1115/1.4027865
[59]	B. Wang, Q. Zhu, S. Li, Stability analysis of discrete-time semi-Markov jump linear systems with time delay, IEEE T. Automat. Control, 68 (2023), 6758–6765. https://doi.org/10.1109/TAC.2023.3240926 doi: 10.1109/TAC.2023.3240926
[60]	F. Li, J. Liu, Y. Yan, J. Rong, J. Yi, A time-variant reliability analysis method based on the stochastic process discretization under random and interval variables, Symmetry, 13 (2021), 568. https://doi.org/10.3390/sym13040568 doi: 10.3390/sym13040568
[61]	H. M. Qian, Y. F. Li, H. Z. Huang, Improved model for computing time-variant reliability based on outcrossing rate, ASCE-ASME J. Risk U. A, 6 (2020), 04020043. https://doi.org/10.1061/AJRUA6.0001090 doi: 10.1061/AJRUA6.0001090
[62]	J. Wang, G. Xu, Y. Li, A. Kareem, AKSE: A novel adaptive Kriging method combining sampling region scheme and error-based stopping criterion for structural reliability analysis, Reliab. Eng. Syst. Safe., 219 (2022), 108214. https://doi.org/10.1016/j.ress.2021.108214 doi: 10.1016/j.ress.2021.108214
[63]	Y. Zhao, L. Wang, H. Wu, H. Chu, C. Yang, Z. Liu, Time‐varying reliability method based on linearized Nataf transform, Qual. Reliab. Eng. Int., 37 (2021), 1922–1938. https://doi.org/10.1002/qre.2836 doi: 10.1002/qre.2836
[64]	H. Liu, J. Cai, Y. S. Ong, An adaptive sampling approach for Kriging metamodeling by maximizing expected prediction error, Comput. Chem. Eng., 106 (2017), 171–182. https://doi.org/10.1016/j.compchemeng.2017.05.025 doi: 10.1016/j.compchemeng.2017.05.025
[65]	S. Yang, D. Meng, Y. Guo, P. Nie, A. M. De. Jesus, A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures, Int. J. Struct. Integr., 14 (2023), 809–826. https://doi.org/10.1108/IJSI-06-2023-0049 doi: 10.1108/IJSI-06-2023-0049
[66]	L. Xin, H. Lai, X. Wang, X. Song, K. Liu, S. Wu, et al., Aerospace structural reliability analysis method based on regular vine copula model with the asymmetric tail correlation, Aerosp. Sci. Technol., 142 (2023), 108670. https://doi.org/10.1016/j.ast.2023.108670 doi: 10.1016/j.ast.2023.108670
[67]	S. Yang, D. Meng, H. Wang, C. Yang, A novel learning function for adaptive surrogate-model-based reliability evaluation, Philos. T. R. Soc. A, 382 (2024), 20220395. https://doi.org/10.1098/rsta.2022.0395 doi: 10.1098/rsta.2022.0395
[68]	D. Meng, S. Yang, H. Yang, A. M. De. Jesus, J. Correia, S. P. Zhu, Intelligent-inspired framework for fatigue reliability evaluation of offshore wind turbine support structures under hybrid uncertainty, Ocean Eng., 307 (2024), 118213. https://doi.org/10.1016/j.oceaneng.2024.118213 doi: 10.1016/j.oceaneng.2024.118213
[69]	Z. Hu, X. Du, First order reliability method for time-variant problems using series expansions, Struct. Multidisc. Optim., 51 (2015), 1–21. https://doi.org/10.1007/s00158-014-1132-9 doi: 10.1007/s00158-014-1132-9
[70]	C. Jiang, X. P. Huang, X. P. Wei, N. Y. Liu, A time-variant reliability analysis method for structural systems based on stochastic process discretization, Int. J. Mech. Mater. Des., 13 (2017), 173–193. https://doi.org/10.1007/s10999-015-9324-z doi: 10.1007/s10999-015-9324-z
[71]	J. N. Fuhg, A. Fau, U. Nackenhorst, State-of-the-art and comparative review of adaptive sampling methods for Kriging, Arch. Computat. Methods Eng., 28 (2021), 2689–2747. https://doi.org/10.1007/s11831-020-09474-6 doi: 10.1007/s11831-020-09474-6
[72]	M. Byrne, M. Dacke, P. Nordström, C. Scholtz, E. Warrant, Visual cues used by ball-rolling dung beetles for orientation, J. Comp. Physiol. A, 189 (2003), 411–418. https://doi.org/10.1007/s00359-003-0415-1 doi: 10.1007/s00359-003-0415-1
[73]	Y. Li, M. Han, Q. Guo, Modified whale optimization algorithm based on tent chaotic mapping and its application in structural optimization, KSCE J. Civ. Eng., 24 (2020), 3703–3713. https://doi.org/10.1007/s12205-020-0504-5 doi: 10.1007/s12205-020-0504-5
[74]	X. Zhang, S. Wen, H. Li, Q. Lu, M. Wu, X. Wang, Chaotic particle swarm optimization algorithm based on Tent mapping, China Mech. Eng., 19 (2008), 0–2094.
[75]	A. G. Hussien, A. A. Heidari, X. Ye, G. Liang, H. Chen, Z. Pan, Boosting whale optimization with evolution strategy and Gaussian random walks: An image segmentation method, Eng. Comput., 39 (2023), 1935–1979. https://doi.org/10.1007/s00366-021-01542-0 doi: 10.1007/s00366-021-01542-0
[76]	Y. Zhang, C. Gong, C. Li, Efficient time-variant reliability analysis through approximating the most probable point trajectory, Struct. Multidisc. Optim., 63 (2021), 289–309. https://doi.org/10.1007/s00158-020-02696-z doi: 10.1007/s00158-020-02696-z
[77]	L. Hawchar, C. P. El. Soueidy, F. Schoefs, Principal component analysis and polynomial chaos expansion for time-variant reliability problems, Reliab. Eng. Syst. Safe., 167 (2017), 406–416. https://doi.org/10.1016/j.ress.2017.06.024 doi: 10.1016/j.ress.2017.06.024
[78]	Z. Hu, X. Du, A sampling approach to extreme value distribution for time-dependent reliability analysis, J. Mech. Des., 135 (2013), 071003. https://doi.org/10.1115/1.4023925 doi: 10.1115/1.4023925
[79]	N. Zou, C. Gong, L. Zhang, Y. Zhang, X. Wang, C. Li, A novel hybrid time-variant reliability analysis method through approximating bound-most-probable point trajectory, Probabilist. Eng. Mech., 75 (2024), 103558. https://doi.org/10.1016/j.probengmech.2023.103558 doi: 10.1016/j.probengmech.2023.103558

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)