Computing the Normalized Laplacian Spectrum and Spanning Tree of the Strong Prism of Octagonal Network

Spectrum analysis and computing have expanded in popularity in recent years as a critical tool for studying and describing the structural properties of molecular graphs. Let O 2 n be the strong prism of an octagonal network O n . In this study, using the normalized Laplacian decomposition theorem, we determine the normalized Laplacian spectrum of O 2 n which consists of the eigenvalues of matrices L A and L S of order 3 n + 1. As applications of the obtained results, the explicit formulae of the degree-Kirchhoﬀ index and the number of spanning trees for O 2 n are on the basis of the relationship between the roots and coeﬃcients. 1.


Introduction
Graphs are a convenient way to depict chemical structures, where atoms are associated with vertices, while chemical bonds are associated with edges. is manifestation carries a wealth of knowledge about the molecule's chemical characteristics.In quantitative structure-activity/property relationship (QSAR/QSPR) studies, one may see that many chemical and physical properties of molecules are closely correlated with graph-theoretical parameters known as topological indices.One such graph-theoretical parameter is the multiplicative degree-Kirchhoff index (see [1]).In statistical physics (see [2]), the enumeration of spanning trees in a graph is a crucial problem.It is interesting to note that the multiplicative degree-Kirchhoff index is closely related to the number of spanning trees in a graph.e normalized Laplacian acts as a link between them.
Let G be an n-vertex simple, undirected, and connected graph with the vertex set of V(G) and an edge set of E(G).
For standard notation and terminology, one may refer to the recent papers (see [3,4]).
e (combinatorial) Laplacian matrix of graph G is specified as L G � D G − A G , where D G is the vertex degree diagonal matrix of order n and A(G) is an adjacency matrix of order n.
e normalized Laplacian is defined by Evidently, L(G) � D(G) − A(G) and L(G) � D(G) − 1/2 L(G)D(G) − 1/2 .As we all know, the normalized Laplacian technique is useful for analyzing the structural features of nonregular graphs.In reality, the interaction between a graph's structural features and its eigenvalues is the focus of spectral graph theory.For more information, see recent articles [5][6][7][8] or the book [9].
Many parameters were used to characterize and describe the structural features of graphs in chemical graph theory.
e Wiener index [10,11] was a well-known distance-based index, as it is known as W(G) �  i<j d ij .Eventually, Gutman [12] defined the Gutman index as follows: ( In accordance with electrical network theory, Klein and Randić [13] presented a new distance function called resistance distance that is denoted as r ij .e resistance distance in electrical networks is between two arbitrary vertices i and j when every edge is replaced by a unit resistor.Klein and Ivanciuc [14] called it the Kirchhoff index, the total sum of resistance distances between each pair of vertices of G, which is Kf(G) �  i<j r ij .Later, the degree-Kirchhoff index was established by Chen and Zhang [1] and denoted by Kf * (G) �  i<j d i d j r ij .
Because of their practical uses in physics, chemistry, and other sciences, the Kirchhoff index and the degree-Kirchhoff index have gained a lot of attention.Klein and Lovász [15,16] separately established that

the eigenvalues of L(G).
According to Chen [17], the degree-Kirchhoff index is, where Since the Kirchhoff index and multiplicative degree-Kirchhoff index have been widely used in the domains of physics, chemistry, and network science.During the previous few decades, many scientists have been working on explicit formulae for the Kirchhoff and degree-Kirchhoff indices of graphs with particular structures, such as cycles [18], complete multipartite graphs [19], generalized phenylene [20], crossed octagonal [21], hexagonal chains [22], pentagonal-quadrilateral network [23], and so on.Other research on the Kirchhoff index and the multiplicative degree-Kirchhoff index of a graph has been published (see [24][25][26][27][28][29][30][31]).In organic chemistry, polyomino systems have received a lot of attention, especially in polycyclic aromatic compounds.Tree-like octagonal networks are condensed into octagonal networks that belong to the polycyclic conjugated hydrocarbons' family.
e octagonal system without any branches is known as a linear octagonal network [32].As shown in Figure 1, a linear octagonal network could also be created from a linear polyomino network by adding additional points to the line according to specified rules.
e strong product between the graphs G and H is denoted by G⊠H, where the vertex set and ab ∈ E G .In particular, the strong product of K 2 and G is known as the strong prism of G. Recently, Li [33] and Ali [34] [34][35][36], we derive an explicit analytical expression for the multiplicative degree-Kirchhoff index and also spanning trees of O 2 n .

Preliminaries
In this section, we start by going over some basic notation and then introduce a suitable technique.Given the square matrix R having order n, we refer to R[i 1 , i 2 , . . ., i k ] as the submatrix of R that results from deleting the i 1 th, i 2 th, . .., i k th columns and rows.Let Φ(R) � det(xI n − R) be the characteristic polynomial of the square matrix R. e labeled vertices of O 2 n are as depicted in Figure 2 and n ) could be represented as a block matrix below: It is simple to verify that en, where Huang et al. obtained the following lemma.2

Main Results
In this section, we are committed to the explicit analytical solution for the multiplicative degree-Kirchhoff index, as well as the spanning tree of O 2 n .In terms of the role of normalized Laplacian L, the following block matrices of n ) are obtained according to equation (8).Journal of Mathematics 3 By equation ( 8), we have a matrix of order 6n + 2: and us, (1/2)L A could be represented by the block matrix below: Let en, where T ′ indicates the transposition of T.

Journal of Mathematics
Proof of Claim 1. Noticing that (−1) 3n− 1 a 3n−1 is equal to the sum of all principal minors of P with 3n − 1 columns and rows, we have where Note that then Z is an empty matrix and let detZ � 1.By equation ( 26), there are different possibilities which can be selected for i and j. erefore, all these cases are classified as follows.

Conclusion
In this study, we consider O 2 n , which is the strong prism of the octagonal network.Using the normalized Laplacian theorems, we have determined the multiplicative degree-Kirchhoff index and the spanning tree of O 2 n .New discoveries, developments, and advancements in research are still required.In the near future, we will be exploring a more complex chemistry network.

Figure 1 :
Figure 1: Graph O n with labeled vertices.
p−1)+2 • r 3(n−p)+1 +  n−1 p�1 r 3p • r 3(n−p) +  n−1 p�0 r 3p+1 • r 3(n−p−1)+2 + 2r 3n .(47)e following forms can also be generated by using the above equations: calculated the resistance distance-based parameters of the strong prism of unique graphs, such as strong prism of S n and L n ⊠K 2 , respectively.Let O 2 n be the strong prism of K 2 and O n , denoted by O 2 n � K 2 ⊠O n , as shown in Figure 2. Obviously, |E(O 2 n )| � 34n + 6 and |V(O 2 n )| � 12n + 4.In this paper, motivated by eigenvalues of P and Q, respectively.