The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal- Quadrilateral Networks

/e normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L8,4 n represent a linear octagonal-quadrilateral network. /en, by identifying the opposite lateral edges of L8,4 n , we get the corresponding Möbius graph MQn(8, 4). In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of MQn(8, 4) can be determined by the eigenvalues of two symmetric quasi-triangular matricesLA andLS of order 4n. Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of MQn(8, 4).


Introduction
It is well established that networks can be represented by graphs. e graphs we consider in this paper are simple, undirected, and connected. Let us first recall some definitions commonly used in graph theory. Suppose G represents a simple undirected graph with |V G | � n and |E G | � m. For more notations, the readers are referred to [1].
Note that D(G) � diag d 1 , d 2 , . . . , d n represents a degree matrix, where d p is the degree of v p . A(G) is the adjacency matrix of G.
e Laplacian matrix of G is L(G) � D(G) − A(G). e (p, q)-entry of the normalized Laplacian matrix is given by As a matter of fact, there are many parameters that can be used to describe the structure and properties of molecular graphs in graph networks. One of the parameters based on resistance distance is defined as the Wiener index [2,3], which is where d ij � d G (v i , v j ) represents the length of the shortest path between two vertices v i and v j in G. e Wiener index is widely used in chemical and mathematical research. For details, see [4][5][6][7]. e parameter of resistance distance was first proposed by Klein and Randic [8] in 1993. It means that if every edge of a graph G is regarded as a unit resistance, then the distance between any two vertices i and j in G is called resistance distance, which is denoted as r ij . Similar to the Wiener index, we give the expression of the Kirchhoff index [9,10] according to the resistance distance, namely, In 2007, Chen and Zhang [11] proposed that the eigenvalues and eigenvectors of normalized Laplacian spectrum can be used to describe the resistance distance, and the concept of Kirchhoff index is put forward. However, it is very difficult to calculate the degree-Kirchhoff index from the complexity division of graphs, so it is important to find the explicit expression of degree-Kirchhoff index. In recent years, many scholars have devoted themselves to the study of Kirchhoff index of various graphs. Huang et al. [12,13] proved the Kirchhoff index of linear hexagonal chains and linear polyomino chains successively. Ma and Bian [14] determined the normalized Laplacians and degree-Kirchhoff index of cylinder phenylene chain. Liu et al. [15] described the normalized Laplacian and degree-Kirchhoff index of linear octagonal-quadrilateral networks. For more excellent results, refer to [16][17][18][19][20][21]. After learning the excellent works of scholars, in this paper, we use the correlation properties of Laplace matrix to calculate the degree-Kirchhoff index and the complexity of Möbius graph of linear octagonal-quadrilateral networks. e investigation of complex graph and network has gone through a spectacular development in the past decades. Especially in organic chemistry, more and more attention has been paid to the application of polyomino in polycyclic aromatic compounds. Many scholars are interested in the study of linear octagonal networks and related molecular graphs. We all know that linear octagonal network is an octagonal system without branch compression. It is constructed by regularly inserting some new points on the straight line of the linear polyomino network. e research on the structure and properties of this kind of natural graph network lays a solid foundation for the advancement of theoretical chemistry, as well as for the development of applied mathematics.
Let L 8,4 n be the linear octagonal-quadrilateral networks, and octagons and quadrilaterals are connected by a common edge, which are depicted in Figure 1.
en, the corresponding Möbius graph MQ 3 (8, 4) of octagonal-quadrilateral networks is obtained by the reverse identification of the opposite edge by L 8,4 n (see Figure 2). Obviously, we can obtain that |V MQ n (8, 4)| � 8n, |E MQ n (8, 4)| � 10n. e rest of the paper will be divided into the following sections. In Section 2, we put forward some basic notations and related lemmas. In Section 3, we determine the normalized Laplacian spectrum of MQ n (8,4). In Section 4, we present Kemeny's constant, the degree-Kirchhoff index, and the complexity of MQ n (8, 4).

Preliminary
In this section, we introduce some common symbols and related calculation methods [1], which are applied to the rest of the article. e characteristic polynomial of matrix R of order n is defined as P R (x) � det(xI − R). It is not difficult to find that π is an automorphism of G, and we can write the product of disjoint 1-cycles and transposition, namely, π � (1)(2), . . . , (m) 1, 1 ′ 2, 2 ′ , . . . , k, k ′ . (4) en, one has |V(G)| � m + 2k, and let v 0 � 1, 2, us, the Laplacian matrix can be expressed in the form of block matrix, that is, where Let P � and then Note that P ′ is the transposition of P, where Lemma 1 (see [12]). Let L(L n )(G), L A (G), L S (G) be determined as above; then, (b) (see [11]). e degree-Kirchhoff index of G is defined as (c) (see [1]). e number of spanning trees of G can also be called the complexity of G. en, the complexity of G is
Next, we calculate some main results of MQ n related to the normalized Laplacian spectrum.

The Degree-Kirchhoff Index and the
Complexity of MQ n (8,4) In this section, we first introduce some theorems, which are obtained by describing the eigenvalues and eigenvectors of normalized Laplacian matrix. en, we obtain Kemeny's constant, the degree-Kirchhoff index, and the complexity of Proof. Let en, we can exactly get that η 1 , η 2 , . . . , η 4n are the roots of the following equation: Based on Vieta's theorem of Before calculating (− 1) 4n− 2 a 4n− 2 and (− 1) 4n− 1 a 4n− 1 , we must determine pth order principal submatrices, R 0 p , R 1 p , R 2 p , and R 3 p , which consist of the first p rows and columns of the following matrices L 0 Journal of Mathematics In this way, we can get four facts. , if p ≡ 2(mod4), Fact 4. For 1 ≤ p ≤ 4n, Proof. of Fact 1. Take r 0 p � detR 0 p , r 1 p � detR 1 p , r 2 p � detR 2 p , and r 3 p � detR 3 p . By a straightforward calculation, one can get the following values (see Table 1).
For 4 ≤ p ≤ 4n − 1, we can get expansion formula of det R 0 p with respect to its last row: en, it is not difficult to obtain that According to the equation of d p in (28), it is evident to see that x 2 − (1/18)x + (1/1296) � 0, and its two roots are (1/36) and (1/36). erefore, d p � (x p + y)(1/36) p is the general solution. en, we can get y � 1 3 , us, we can obtain e result is obtained as desired.
Proof. of Claim 1. Since (− 1) 4n− 1 a 4n− 1 is the total of all the principal minors of order 4n − 1 of L A , we have where Journal of Mathematics 4n p�4,p≡0(mod4)  (32) Similarly, by Facts 1-4, we can get Hence, according to the above results, we have Proof. of Claim 2. It is not hard to see that (− 1) 4n− 2 a 4n− 2 is the total of those principal minors L A , which have (4n − 2) rows and columns. us, we have By equation (35), it can be seen that the change of i and j values will lead to different detL A [p, q] results. erefore, we will choose different p and q to list the following equations: By Facts 1-4, we can compute the following results. Case 1. (37)

Journal of Mathematics 11
Case 2.
(57) en, we can exactly get that φ 1 , φ 2 , . . . , φ 4n are the roots of the following equation: Based on Vieta's theorem of P L S (x), one has Before calculating (− 1) 4n− 1 b 4n− 1 and detL S , we must determine ith order principal submatrices S 0 q , S 1 q , S 2 q , and S 3 q , which consist of the first q rows and columns of the matrices L 0 S , L 1 S , L 2 S , and L 3 S , respectively, q � 1, 2, . . . , 4n. Let