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The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks
Let be the linear heptagonal networks with heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of , we utilize the method of decompositions. Thus, the Laplacian spectrum of is created by eigenvalues of a pair of matrices: and of order numbers and , respectively. On the basis of the roots and coefficients of their characteristic polynomials of and , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of .
With the discovery of carbon nanotubes, there has been a practical interest in the study of concerning structures based on hexagon networks. Especially for a class of linear hexagonal networks, Yang  and Huang et al. , respectively, explored Kirchhoff index and another form multiplicative degree-Kirchhoff index of linear hexagonal networks. Years later, Peng et al.  had studied the Kirchhoff index and related spanning trees for the linear phenylenes which are constructed from polyominoes and hexagons. Inspired by , the multiplicative version of degree-Kirchhoff index and its numbers of spanning trees of the generalized phenylenes are given. We refer the associated work [5–9], and the notation we use in this paper follows that in .
Tips of carbon nanotubes are closed by pentagons. Heptagons, which generate negative curvatures, can also be introduced with those pentagons . Additionally, there are abundant practical applications of the heptagons in nature aspects; a remarkable example is that cacti are the most common plants with heptagons in natural structures. In our work, we concern a class of linear heptagonal networks inspired by these linear hexagonal networks (see Figure 1). Evidently, we see that is an automorphism .
Before proceeding, we shall review some terminologies for the chemical graph theory. Since the topological indices are closely related to the physical and chemical properties of the corresponding molecular graph, the calculation of various topological indices is the core subject of chemical graph theory [11–20]. At this point, we slightly observe some properties of the Kirchhoff index. Klein and Randić in  defined a summation of all resistance distances between pairs of vertices from a graph as its Kirchhoff index , namely,where is its resistance distance between two vertices and [22–26].
Lately, one finds that Kirchhoff index of a graph is closely related to the Laplacian eigenvalue. That is,in which are Laplacian eigenvalues of .
The organization of this paper is as follows. In Section 2, some necessary notations of the block matrices are reviewed. Then, we apply automorphism and some lemmas into . The Laplacian matrix can be considered to have block matrices, and we can obtain the results of these block matrices. Finally, we use transposition of matrices to get and . In Section 3, we first study those explicit formulas of Kirchhoff index for . Connecting these eigenvalues of and , we can obtain the Laplacian spectrum of . In the end, we obtain the number of spanning trees of .
Before embarking on the proofs of our main results, we observe a special method that is the basis of our work. These equations that we describe can be found in [5,13].
For an by matrix , denote by this submatrix of , and create by the deletion of the -th, …, -th rows and the columns. In the following part, is this characteristic polynomial of .
Label these vertex sets of by , , and . So, the Laplacian matrix of will be block matrices.
Note that , and .
Consider the unitary transformation of ; then,where is the transposition of . Obviously,
Furthermore, we present those matrices introduced above in the following. According to the structure of Figure 1, one obtains .where and ; else, .
In addition, one getswhere for ; else, it is 0.
According to equation (2), one haswhere , with and for , for . Also, for and with .
Also,where for is odd and for is even; with , and with .
Next, we list some lemmas that may be useful throughout our main proof.
Lemma 1 (see ). Suppose that are mentioned before. We have
Based on the properties of these spanning trees and Laplacian eigenvalues, we have the following.
Lemma 2 (see ). Let be connected graph with and be the complexity in . We have
Lemma 3 (see ). Let , and be the square matrices with by , by , by , and by matrices, respectively. We havewhere is invertible and are denoted as Schur complements of .
3. Kirchhoff Index and the Number of Spanning Trees of
So far, our first main result provides the exact formula to the Kirchhoff index of . Together with their eigenvalues of and , it is not hard to use Lemma 1 and obtain Laplacian spectrum of . Our second main result is to furnish the complete information of the number of spanning trees of , which contributed to degree products and their eigenvalues. Suppose and are, respectively, the roots of and . By equation (2), one has the following.
Lemma 4. Let be the linear heptagonal networks. We have
Obviously, we only need to compute the eigenvalues of and . Thus, the formulas of and are given as follows.
Lemma 5. Assume that are the eigenvalues of . One has
Proof. It is known thatwith . Then, we find that satisfy the equationIn the line with Vieta theorem, it gives
Proposition 1. .
Proof. Because is the summation of these principal minors from , that has rows and columns, it is easy to see thatIn view of equation (2), we know that can be chosen from the matrix or . Hence, we consider the cases below.
(1) Case 1. For the case of , is created by dropping the -th row of matrices and and also the related columns of and . Let , and be these blocks. By using Lemma 3 and fundamental operations, one knows thatwhere , of which for , but , . Also, for and for . By routine calculations, we have(2) Case 2. For the case of , we say is deduced by deleting the -th row of and and the corresponding column of and . Denote , and as desired blocks. By Lemma 3 and fundamental operations, we obtain thatwhere . Set with order , of which for . Also, , for and with . By routine calculations, we have
Combining equations (4)–(6), we have
So, the proof of Proposition 1 is finished.
Next, we will focus on recursive calculations for .
Proof. We know that is the summation of their principal minors of giving rows and columns. We can obtainFor the result of Proposition 1, we will consider the following cases.
(1) Case 1. For the case of , we see that is obtained by dropping and rows of the identity matrices and and the corresponding columns of and . Denote , and as desired blocks. Based on Lemma 3 and elementary operations, we havewhere , of which for and ; also, , . Then, with , with . By basic calculations, we have(2) Case 2. For the case of , we say that is created by dropping -th and -th rows of and and the corresponding columns of and . Denote , and as desired blocks. In terms of Lemma 3 and fundamental operations, we obtainwhere . By recursive calculations, we obtain(3) Case 3. We consider the case of , . In other words, we have that is created by dropping rows of the identity matrices and and -th rows of and , respectively. For further consideration, we write them as , , and . Denote , and as desired blocks. By the methods of Lemma 3 and fundamental operations, we havewhere. By an explicit calculation, we get
Considering equations (7)–(10), one acquires
Corollary 1. Suppose that are the roots of . Then,Let . By using the relation between the coefficients and roots of , it is not hard to see that
On the other hand, we will consider -th ordered principal submatrix, , which is found by giving first rows and columns of . Let .
Fact 1. For ,Proof. We will compute and obtain that . For , we can extend with its last row as follows:For , let , and for , let . Then we have, together with (12), , and for ,From the first equation in (13), one has ; therefore, . Replacing and into the second equation in (13) yields , . Similarly, we obtain , . Thus, meets the recurrence relationSo, we see that the characteristic equation of (40) is , whose roots are , , , and .
Assume that the general solution of (14) isCombining the initial condition in (14), the system of equations is given:The unique solution of this system of equation is , , , and . Therefore, Fact 1 is deduced by (15).
By using the expansion formula, one can deduce with its last row below .Combining it with Fact 1, the following is obtained.
Fact 2. .
Now we are ready to determine . We can denote this diagonal of by .
Proof. Since is the summation of these principal minors from with size by , we havewhereBecause permutating rows and related columns in square matrices keeps its determinant unchanged, combining it with , we will get . Therefore,By Fact 1, we have that andWe can use a similar routine to obtain
Lemma 6. For graph ,where , .
The Kirchhoff indices of linear heptagonal networks from to are denoted in Table 1.
Lemma 7. For graph ,
Proof. By Lemma 2, we have Note thatThen,Next, the number of spanning trees in linear heptagonal networks from to is given in Table 2.
The figures, tables, and other data used to support this study are included within the article.
An online preprint of this study has been published elsewhere .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported in part by the Anhui Provincial Natural Science Foundation under grant no. 2008085J01 and Natural Science Fund of Education Department of Anhui Province under grant no. KJ2020A0478.
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