On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

: The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2 n squares, denoted by L 6,4,4 n . In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of L 6,4,4 n consisting of the eigenvalues of symmetric tri-diagonal matrices L A and L S of order 4 n + 1. As an application, the signiﬁcant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefﬁcients and roots.


Introduction and Preliminaries
In this paper, only simple, undirected, and finite graphs are considered. Assume that G = (V G , E G ) is a graph with vertex set V G and edge set E G . The order of G is denoted by n = |V G | and the size of G is denoted by m = |E G |. The fundamental expressions and methodologies of graph theory have been used (see [1]). Assume that G is a graph of n vertices, adjacency matrix A G = [a ij ] n×n is a matrix, such that a ij equals 1 if vertices v i and v j are adjacent and zero for otherwise. Assume that D G = diag(d v 1 , . . . , d v n ) is the vertex degree diagonal matrix of order n, where d v i is the degree of v i , 1 ≤ i ≤ n. Then L G = D G − A G is called the (combinational) Laplacian matrix of G.
The traditional concept of distance between vertices v i and v j , is the length of the shortest path obtained by joining these vertices of graph G and that is denoted by d ij = d G (v i , v j ). In graph theory, distance is also an essential invariant from which distance based parameters are obtained. In [2], a well-known distance based parameter named as Wiener index and denoted by W G is introduced for the first time. This parameter is obtained by adding the distances between every pair of vertices in G, that is, W G = ∑ {v i ,v j }⊆V G d ij . Eventually, Gutman [3] also developed the Gutman index, which is a weighted variant of the Wiener index and defined as Gut(G) = ∑ {v i ,v j }⊆V G d i d j d ij .
In [4], Klein and Randić suggested a novel distance function r ij for a graph depending upon the electrical network theory, which is called the resistance distance. The Kirchhoff index was generated by Klein and Ivanciuc [5] and Klein and Randi [4] based on the resistance distance parameter and is denoted by K f (G) = ∑ {v i ,v j }⊆V G r ij .
Gutman and Mohar [6] established a result separately for the Kirchhoff index that is represented as, where µ 1 < µ 2 ≤ · · · ≤ µ n with µ 1 = 0 and n ≥ 2 are the eigenvalues of L G . In [7], Chung proposed the normalized Laplacian, denoted by L G , that is, G As a result, it is easy to understand that where d i (d j ) be the degree of the vertex v i (v j ) and (L G ) ij show the (i, j)-th values of L G . In 2007, Chen and Zhang [8] presented a new index based on normalized Laplacian, which is called the multiplicative degree-Kirchhoff index and defined it as, . Moreover, the Kirchhoff index has gained much attention due to its widespread use in chemistry, physics, mathematics, and theoretical computer science. Many scientists have previously proposed new closed formulae of Kirchhoff and multiplicative degree-Kirchhoff indies, along with linear polyomino chains [9], circulant graphs [10], quadrilateral graphs [11], cycles [12], composite graphs [13], and so on. There are several results on the degree-Kirchhoff index and normalized Laplacian (see [14][15][16][17][18][19][20][21][22][23][24][25]).
Due to the widely applications, the molecular graphs phenylenes, pentagonal, hexagonal, and octagonal networks have attracted strong interests of chemists, mathematicians, and engineering. Phenylenes are two connected graphs with structure-property that each of their cells (or interior faces) has become a hexagon or square with the same edge length. Phenylene systems are essential in theoretical chemistry because they can reflect a hydrocarbon in nature. In [26,27], Gutman et al. addressed the phenylene enumeration problem using the Kekulé structure. Later, Pavlović and Gutman [28] have solved the problem of evaluating the Wiener index of phenylenes. Chen and Zhang [29] calculated the expected value of Wiener index (or the number of perfect matches) of the unique phenylene chain using a specific analytical expression. Recently, Geng and Lei [30] explicitly calculated the Kirchhoff index and spanning tree of phenylene chain.
In this article, by using the decomposition theorem for the normalized Laplacian characteristic polynomial, the multiplicative degree-Kirchhoff index, as well as spanning trees of L 6,4,4 n are given explicit closed-form formulas.
Lemma 2 (see [8]). Assume that G is an n vertex graph having size m, then K f * (G) = 2m Lemma 3 (see [7]). Assume that G is a connected graph having size m, then τ(

Main Results
In this section, the normalized Laplacian eigenvalues of L n are first obtained. Next, we resolve to discover the calculation for multiplicative degree-Kirchhoff index (or spanning tree) of L n . Given a square matrix A with order n, we will write to A[{i 1 , i 2 , . . . , i k }] as the submatrix generated by eliminating the i 1 th, i 2 th, . . . , i k th columns and rows of A. According to Equation (3), the block matrices could be represented as follows. and .
where α i , (i = 2, 3, . . . , 4n + 1) (α i > 2) and β i , (i = 1, 2, . . . , 4n + 1) are eigenvalues of L A and L S , respectively. According to the relationship between the coefficients and roots of Ψ(L A ) (respectively. Ψ(L S )), standard formulae of ∑ 4n+1 Lemma 5. Suppose that 0 = α 1 < α 2 ≤ · · · ≤ α 4n+1 to be described as above. Then Then α 2 , α 3 , . . . , α 4n+1 are roots satisfies the equation below: are roots satisfies the equation below: Hence, by Vieta's theorem (see [31], p. 81), we obtain For the sake of convenience, we take S i to the i-th order principal sub-matrix of L A , yield by the first ith columns and rows, i = 1, 2, . . . , 4n. Let s i = det S i . Then These formulas in general form could be derived by a straightforward calculation as shown below. Case 1. Suppose s 4i , s 4i+1 , s 4i+2 and s 4i+3 are defined as above. We have In contrast, we take T i to the i-th order principal sub-matrix of L A , built into the last ith columns and rows, i = 1, 2, . . . , 4n. Let t i = det T i . Then These formulas in general form could be derived by a straightforward calculation as shown below. Case 2. Suppose t 4i , t 4i+1 , t 4i+2 and t 4i+3 are defined as above. We have When case 1 and case 2 are combined, the following fact can be deduced.
where Combining Case 1, Case 2 and Equation (6), we find as desired. where In view of Equation (8), we know that det X will change as a result of the various i and j options. Hence, we will discuss by separating the cases below. Case 1. Let i = 4k and j = 4l, for 1 ≤ i < j ≤ 4n + 1. Therefore, 1 ≤ k < l ≤ n. In this case, X is the square matrix of order (4l − 4k − 1).
In this case, X is the square matrix of order (4l − 4k).
In this case, X is the square matrix of order (4l − 4k + 1).
In this case, X is the square matrix of order (4l − 4k + 2).
In this case, X is the square matrix of order (4l − 4k − 1).
In this case, X is the square matrix of order (4l − 4k − 2).
In this case, X is the square matrix of order (4l − 4k − 1).
In this case, X is the square matrix of order (4l − 4k).
In this case, X is the square matrix of order (4l − 4k − 3).
In this case, X is the square matrix of order (4l − 4k − 2).

Conclusions
In this article, the molecular graph L 6,4,4 n of generalized phenylene containing 2n squares and n hexagons is considered. According to the decomposition theorem of normalized Laplacian polynomial, we achieve expressive formulas for the degree-Kirchhoff index and spanning tree of L 6,4,4 n .
Author Contributions: Writing-original draft preparation, U.A., H.R. and Y.A.; writing-review and editing, U.A., H.R. and Y.A. All authors contributed equally to this manuscript. All authors have read and agreed to the published version of the manuscript.