On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

: A novel distance function named resistance distance was introduced on the basis of electrical network theory. The resistance distance between any two vertices u and v in graph G is deﬁned to be the effective resistance between them when unit resistors are placed on every edge of G . The degree-Kirchhoff index of G is the sum of the product of resistance distances and degrees between all pairs of vertices of G . In this article, according to the decomposition theorem for the normalized Laplacian polynomial of the linear pentagonal derivation chain QP n , the normalize Laplacian spectrum of QP n is determined. Combining with the relationship between the roots and the coefﬁcients of the characteristic polynomials, the explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of QP n can be obtained, respectively. Moreover, we also obtain the Gutman index of QP n and we discovery that the degree-Kirchhoff index of QP n is almost half of its Gutman index.


Introduction
Throughout this paper, we handle a simple, finite, and undirected graph.Let G = (V(G), E(G)) be a graph with vertex set V(G) = {v 1 , v 2 , . . ., v n } and edge set E(G).For v i ∈ V(G), let N G (v i ) be the set of neighbors of v i in G.In particular, d i = |N G (v i )| is the degree of v i in G.The adjacency matrix of G, written as A(G), is an n × n matrix whose (i, j)-entry is 1 if v i v j ∈ E(G) or 0 otherwise.The Laplacian matrix L(G) = D(G) − A(G), where D(G) = diag(d 1 , d 2 , . . ., d n ) is the diagonal matrix of G whose diagonal entry d i is the degree of v i for 1 ≤ i ≤ n.
The normalized Laplacian matrix [1] of a graph G, L(G), is defined to be with the convention that D(G) −1 (i, i) = 0 if d i = 0. Since the normalized Laplacian matrix is consistent with the eigenvalues in spectral geometry and random walks [1], it has attracted more and more researchers' attention.From the definition of L(G), it is easy to obtain that: For an n × n matrix M, we denote the characteristic polynomial det(xI n − M) of M by Φ M (x), where I n is the identity matrix of order n.In particular, for a graph G, Φ L(G) (x) (respectively, Φ L(G) (x)) is the Laplacian (respectively, normalized Laplacian) characteristic polynomial of G, and its roots are the Laplacian (respectively, normalized Laplacian) eigenvalues of G.The collection of eigenvalues of L(G) (respectively, L(G)) together with their multiplicities are called the L-spectrum (respectively, L-spectrum) of G.
For a graph G, the distance between vertices v i and v j on G is defined as the length of the shortest path between the two vertices, denoted d ij .One famous distance based parameter called the Wiener index [2], which is defined as the sum of the distances between all the vertices on the graph, was given by W(G) = ∑ i<j d ij .For more studies on the Wiener index, one may be referred to [3][4][5][6][7][8].In 1994, Gutman presented an index based on degree and distance of vertex, Gutman index [9], which is Gut(G) = ∑ i<j d i d j d ij .He also showed that when G is an n-order tree, the close relationship between the Wiener index and the Gutman index is Gut(G) = 4W(G) − (2n − 1)(n − 1).
Based on electrical network theory, Klein and Randić [10] proposed a novel distance function named resistance distance.Let G be a connected graph, and the resistance distance between vertices v i and v j , denoted by r ij , is defined as the effective resistance distance between vertices v i and v j in the electrical network obtained by replacing each edge in G with a unit resistance.The resistance distance is a better indicator of the connection between two vertices than the distance.In fact, the resistance distance parameter reflects the intrinsic properties of the graph and has many applications in chemistry [11,12].
One famous parameter called the Kirchhoff index [10], defined as the sum of resistance distances in a simple connected graph, was given by K f (G) = ∑ i<j r ij .In 1993, Klein and Randić [10] proved that r ij ≤ d ij and K f (G) ≤ W(G) with equality if and only if G is a tree.The intrinsic correlation between the Kirchhoff index and the Laplacian eigenvalues of graph G is shown, independently, by Gutman and Mohar [13] and Zhu et al. [14] as where n is the number of vertices of the graph G and 0 As an analogue to the Gutman index, Chen and Zhang [15] presented another graph parameter, the degree-Kirchhoff index K f * (G) = ∑ i<j d i d j r ij .Meanwhile, authors [15] proved that the degree-Kirchhoff index is closely related to the corresponding normalized Laplacian spectrum.Many researchers devote themselves to the study of normalized Laplacian spectrum and the degree-Kirchhoff index of some classes of graphs.One may be referred to those in [16][17][18][19][20][21][22].
As a structured descriptor of chemical molecular graphs, the topological index can reflect some structural characteristics of compounds.Like Kirchhoff index, degree-Kirchhoff index is also a topological index.Unfortunately, it is difficult to compute resistant distance and degree-Kirchhoff index in a graph from their computational complexity.Therefore, it is necessary to find a explicit closed-form formulas for the degree-Kirchhoff index.In fact, the degree-Kirchhoff index is difficult to calculate for general graphs, but it is computable for some graphs with good periodicity and good symmetry.Huang et al. studied the degree-Kirchhoff index of some graphs with a good structure, such as linear polyomino chain [23] and linear hexagonal chain [24].In addition, there are also some studies on the normalized Laplacian spectrum and the degree-Kirchhoff index of phenylene chains [25,26].
The number of spanning trees of a graph (network) is an important quantity to evaluate the reliability of the graph [27].Therefore, studying the number of spanning trees of graphs has a very important theoretical and practical significance.
Hexagonal systems are very important in theoretical chemistry because they are natural graphical representations of benzene molecular structures.In recent years, researchers have worked to study the topological index of hexagonal systems [4,28].The linear pentag-onal derivation chain studied in this paper is related to the hexagonal systems.A linear pentagonal chain of length n, denoted by P n , is made up of 2n pentagons, where every two pentagons with two sides can be seen as a hexagon with one vertex and two sides.Then the linear pentagonal derivation chain, denoted by QP n , is the graph obtained by attaching four-membered rings to each hexagon composed of two pentagons of P n , as showed in  The explicit closed-form formulas for Kirchhoff index and the number of spanning trees of the linear pentagonal derivation chain QP n have been derived from the Laplacian spectrum [29].Motivated by the above works, we consider the degree-Kirchhoff index and the number of spanning trees of linear pentagonal derivation chain in terms of the normalized Laplacian spectrum.Different from the method in [29], in this paper, we solve the number of spanning trees according to the normalized Laplacian spectrum, which gives a new way for calculating the number of spanning trees of QP n .
In this article, according to the decomposition theorem for the normalized Laplacian polynomial of the linear pentagonal derivation chain QP n , the normalized Laplacian spectrum of QP n is determined.Combining with the relationship between the roots and the coefficients of the characteristic polynomials, the explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of QP n can be obtained, respectively.Meanwhile, we also get the Gutman index of QP n .For a general graph G, the ratio . However, we are surprised to discovery that for QP n both (based on our obtained results) as n → ∞.

Preliminaries
In this section, we will give some notations and terminologies and some known results that will be used in our following section.
An automorphism of G is a permutation π of V(G), with the property that v i v j is an edge of G if and only if π(v i )π(v j ) is an edge of G.
Suppose we mark the vertices of QP n as shown in Figure 1 and denote is an automorphism of QP n .For convenience, we abbreviate L(QP n ) to L. By a suitable arrangement of vertices in QP n , the normalized Laplacian matrix L can be written as the following block matrix where L V ij is the submatrix composed by rows corresponding to vertices in V i and columns corresponding to vertices in V j for i, j = 0, 1, 2.

Let
be the block matrix so that the blocks have the same dimension as the corresponding blocks in L. Note that From the unitary transformation TLT, we obtain where According to the above analysis process, Huang et al. [24] derived the decomposition theorem of normalized Laplacian characteristic polynomial of G below.

Lemma 1 ([24]
). Suppose L, L A and L S are defined as above.Then the normalized Laplacian characteristic polynomial of QP n is as follows where Φ L (x), Φ L A (x) and Φ L S (x) are characteristic polynomials of L, L A and L S , respectively.

Lemma 2 ([30]
).Let M 1 , M 2 , M 3 , M 4 be respectively p × p, p × q, q × p, q × q matrices with M 1 and M 4 being invertible.Then Lemma 3. Suppose G is a connected graph of order n with m edges, and

The Normalized Laplacian Spectrum of QP n
In this part, from Lemma 1, we first derive the normalized Laplacian eigenvalues of linear pentagonal derivation chain QP n .Then we present a complete description of the sum of the normalized Laplacian eigenvalues' reciprocals and the product of the normalized Laplacian eigenvalues which will be used in getting the degree-Kirchhoff index and the number of spanning trees of QP n , respectively.
Given an n × n square matrix M, then we will use M[i, j, • • • , k] to denote the submatrix obtained by deleting the i-th, j-th, • • • , k-th rows and corresponding columns of M. In view of (1), L V 00 , L V 01 , L V 12 and L V 11 are given as follows: (see ( 2)), we have , and .
It is easy to see that the normalized Laplacian spectrum of QP n consists of eigenvalues of L A and L S from Lemma 1. Now, suppose that the eigenvalues of L A and L S are, respectively, denoted by [1]).Hence, the eigenvalues of L(QP n ) are nonnegative.That is to say, L A and L S are positive semi-definite.And then, it is not difficult to verify that α 0 = 0, α i > 0 (i = 1, 2, . . ., 4n) and β j > 0 (j = 1, 2, . . ., 3n + 1).

Degree-Kirchhoff Index and the Number of Spanning Trees of QP n
In this section, we first introduce the following lemma which is a direct result of Lemma 3(i).Note that |E(QP n )| = 10n + 1.
Lemma 4. Suppose QP n is a linear pentagonal derivation chain with length n.Then we have Proof.According to the relationship between the roots and coefficients of In the subsequent of this part, it suffices to determine a 4n and −a 4n−1 in Equation (3), respectively.
Proof.One can see that the number a 4n (= (−1) 4n a 4n ) is the sum of the determinants obtained by deleting the i-th row and corresponding column of L A for i = 1, 2, . . ., 4n + 1 (see also in [32]), that is According to the structure of L A (see details in (2)), deleting the i-th row and corresponding column of L A is equivalent to deleting the i-th row and corresponding column of I n , the i-th row of √ 2L V 01 and the i-th column of √ 2L V 10 .We mark the resulting blocks of L A [i], by I n−1 , B (n−1)×(3n+1) , B T (n−1)×(3n+1) , C (3n+1)×(3n+1) , respectively.Then applying Lemma 2 to the resulting matrix, one has where , and there's only one 1 on the diagonal in the (3i (5) In this case, according to the structure of L A , deleting the i-th row and corresponding column of L A is equal to deleting the (i − n)-th row and corresponding column of Expressing the resulting blocks, respectively, as I n , B n×3n , B T n×3n , C 3n×3n .Then by Lemma 2, we obtain where and the E, F are as follows: .
Proof.One can see that −a 4n−1 (= (−1) 4n−1 a 4n−1 ) is the sum of the determinants of the resulting matrix by deleting the i-th row, i-th column and the j-th row, j-th column for some Case 1. 1 ≤ i < j ≤ n.In this case, deleting the i-th and j-th rows and corresponding columns of L A is to deleting the i-th and j-th rows and corresponding columns of I n , the i-th and j-th rows of √ 2L V 01 and the i-th and j-th columns of √ 2L V 10 .Denote the resulting blocks, respectively, as I n−2 , B (n−2)×(3n+1) , B T (n−2)×(3n+1) and C (3n+1)×(3n+1) and apply Lemma 2 to the resulting matrix.Then we have where , and there exists one 1 on the diagonal in the (3i − 1)-th and (3j − 1)-th rows of C − B T B for 1 ≤ i < j ≤ n, respectively .By a direct computing, we have Case 2. n + 1 ≤ i < j ≤ 4n + 1.In this case, deleting the i-th and j-th rows and corresponding columns of L A is to deleting the (i − n)-th and (j − n)-th rows and corresponding columns of L V 11 + L V 12 , the (i − n)-th and (j − n)-th columns of √ 2L V 01 and the (i − n)-th and (j − n)-th rows of √ 2L V 10 .Similarly, denote the resulting blocks, respectively, as C (3n−1)×(3n−1) , B n×(3n−1) , B T n×(3n−1) and I n .Then by Lemma 2 to the resulting matrix, we have where , and the E, F, G are as follows: .
By a direct calculation, we have By using a similar method, deleting the i-th and j-th rows and corresponding columns of L A is to deleting the i-th row and i-th column of I n , the (j − n)-th row and (j − n)-th column of L V 11 + L V 12 , the i-th row and (j − n)-th column of √ 2L V 01 and the (j − n)-th row and i-th column of √ 2L V 10 .We denote the resulting blocks, respectively, as I (n−1) , C 3n×3n , B (n−1)×3n and B T (n−1)×3n and apply Lemma 2 to the resulting matrix.Then we get and there is only one 1 on the diagonal in the (3i and there is only one 1 on the diagonal in the (3i where , and there is only one 1 in the (3i − 1)-th row of E, or where Combining with ( 7)- (10), we obtain Finally, substituting Claims 1 and 2 into (3), Lemma 5 holds directly.
are the roots of the Φ L S (x) = 0. Applying Vieta's Formulas [31], we get 3n+1 ∑ j=1 1 In order to determine (−1) 3n b 3n and det L S in (11), we consider the k order principal submatrix W k consisting of the first k rows and the first k columns of L S , k = 1, 2, . . ., 3n + 1.Put w k := det W k .Let's prove the following fact first.
Proof.By a direct calculation, we obtain that w 1 = 3 2 , w 2 = 4 3 , w 3 = 29 18 , w 4 = 54 27 , w 5 = 295 162 , and w 6 = 536 243 , expanding det W k with regard to its last row, we have According to Theorem 1, we can have the degree-Kirchhoff indices of linear pentagonal derivation chains from QP 1 to QP 40 , as shown in Table 1.
Based on Claims 1, 3 and Lemma 3, we can get the same results as the Theorem 3 [29], which further proves that the result of our calculation (Theorem 2) is correct.Theorem 2. Let QP n denote a linear pentagonal derivation chain with length n.Then

A Relation between the Gutman Index and Degree-Kirchhoff of QP n
At the end of this paper, we calculate the Gutman index and show that the degree-Kirchhoff index of QP n is about half of its Gutman index.Theorem 3. Let QP n denote a linear pentagonal derivation chain with length n.Then Gut(QP n ) = 200n 3 + 181n 2 + 31n + 1.
Proof.Let the vertices of QP n be labeled as in Figure 1.Recall that Gut(G) = ∑ i<j d i d j d ij .Therefore, we evaluated d i d j d ij for all vertices, and then we summed them and divided by two.First, compute d i d j d ij for each type of vertices separately and the expression of each type of vertices are as follows: Fixed the vertices 1 or 1 of QP n : Fixed the vertices 2 or 2 of QP n : Fixed the vertices 3l or 3l (1 ≤ l ≤ n) of QP n :  Fixed the vertices 3n + 1 or 3n + 1 of QP n : Fixed the vertex 1 • of QP n : Fixed the vertices l • (2 ≤ l ≤ n − 1) of QP n : Fixed the vertex n • of QP n :

Figure 1 .
Figure 1.The linear pentagonal derivation chain QP n .
eigenvalues of L S .Then .Proof.Similarly, for Φ L S

Table 1 .
The degree-Kirchhoff indices of linear pentagonal derivation chains from QP 1 to QP 40 .