Abstract
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 be the strong prism of an octagonal network . In this study, using the normalized Laplacian decomposition theorem, we determine the normalized Laplacian spectrum of which consists of the eigenvalues of matrices and of order . As applications of the obtained results, the explicit formulae of the degree-Kirchhoff index and the number of spanning trees for 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. This 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. The normalized Laplacian acts as a link between them.
Let be an -vertex simple, undirected, and connected graph with the vertex set of and an edge set of . For standard notation and terminology, one may refer to the recent papers (see [3, 4]). The (combinatorial) Laplacian matrix of graph is specified as , where is the vertex degree diagonal matrix of order and is an adjacency matrix of order .
The normalized Laplacian is defined by
Evidently, and . 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 [5–8] or the book [9].
Many parameters were used to characterize and describe the structural features of graphs in chemical graph theory. The Wiener index [10, 11] was a well-known distance-based index, as it is known as . 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 . The resistance distance in electrical networks is between two arbitrary vertices and 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 , which is . Later, the degree-Kirchhoff index was established by Chen and Zhang [1] and denoted by .
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 thatwhere are the eigenvalues of . According to Chen [17], the degree-Kirchhoff index is,where are the eigenvalues of .
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 [24–31]). 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. The 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.
The strong product between the graphs and is denoted by , where the vertex set is and is an edge of if and is adjacent to in or and is adjacent to in or and . In particular, the strong product of and is known as the strong prism of . Recently, Li [33] and Ali [34] calculated the resistance distance-based parameters of the strong prism of unique graphs, such as strong prism of and , respectively. Let be the strong prism of and , denoted by , as shown in Figure 2. Obviously, and .
In this paper, motivated by [34–36], we derive an explicit analytical expression for the multiplicative degree-Kirchhoff index and also spanning trees of .
2. Preliminaries
In this section, we start by going over some basic notation and then introduce a suitable technique. Given the square matrix having order , we refer to as the submatrix of that results from deleting the th, th, …, th columns and rows. Let be the characteristic polynomial of the square matrix . The labeled vertices of are as depicted in Figure 2 and and . The normalized Laplacian matrix could be represented as a block matrix below:
It is simple to verify that and .
Let
Then,where
Huang et al. obtained the following lemma.
Lemma 1 (see [8]). Let be a graph and let and be as described above. Then, we have .
Lemma 2 (see [1]). Let be the eigenvalues of ; then, the degree-Kirchhoff index can also be written as .
Lemma 3 (see [17]). Let be -vertex connected graph of size ; then, the spanning trees is .
3. 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 . In terms of the role of normalized Laplacian , the following block matrices of and are obtained according to equation (8).
By equation (8), we have a matrix of order :and , a diagonal matrix with order .
The normalized Laplacian spectrum of is constructed by the eigenvalues of and , according to Lemma 1. Given the fact that is just a diagonal matrix of order , it is obvious that with multiplicity and with multiplicity are the eigenvalues of .
Let
Thus, could be represented by the block matrix below:
Let
Then,where indicates the transposition of . Let and . Then,
By Lemma 1, it is simple to verify that the eigenvalues of consist of those of and . Suppose that the eigenvalues of and are denoted by and with and , respectively. Then, the eigenvalues of are and .where and are eigenvalues of and , respectively.
Lemma 4. Suppose that is the strong product of octagonal network. Then,
On the basis of the relation between the coefficients and roots of (resp. ), the formulae of (resp. ) are obtained in the next lemmas.
Lemma 5. Suppose that are described as above. Then,
Suppose that Then, satisfy the equation below:so satisfy the equation below:
Hence, by Vieta’s theorem, we obtain
For the sake of convenience, consider of , which is the th order principal submatrix generated by the first columns and rows, . Let . Then,
By explicit calculation, these general formulae can be obtained as follows:
The structure and determinant of matrix are preserved by a permutation similarity transformation of a square matrix, and one gets . We haveas desired.
Claim 1. .
Proof . of Claim 1. Noticing that is equal to the sum of all principal minors of with columns and rows, we havewhereNote that
Remark 1. If , then is an empty matrix and let . By equation (26), there are different possibilities which can be selected for and . Therefore, all these cases are classified as follows.
Case 1. Let and , for . So, :
Case 2. Let and , for . So, :
Case 3. Let and , for . So, :
Case 4. Let and , for . So, :
Case 5. Let and , for . So, :
Case 6. Let and , for . So, :
Case 7. Let and , for . So, :
Case 8. Let and , for . So, :
Case 9. Let and , for . So, :Combining these results with equation (26) and Cases 1–9 yieldswhereSubstituting , , and in Equation (36), we get Claim 2.
Also, we can get Lemma 5 by combining Claims 1 and 2.
Lemma 6. Let be the eigenvalues of as above. Then,where and .
Proof . Suppose that .
So, satisfy the equation below:By Vieta’s theorem, we obtainIn order to find and in (40), consider of , which is the th order principal submatrix generated by the first columns and rows, . Let . Then, , , , , , , andSimilar to the method used as described above, we have
Fact 1. .
Proof . Fact 1. Expanding along the last row, we have
Fact 2.
Proof . of Fact 2. Noting that is the summation of all principal minors of with columns and rows, we havewhereThe structure and determinant of matrix are preserved by a permutation similarity transformation of a square matrix, and one gets . In line with Equation (45), we haveThe following forms can also be generated by using the above equations:We can obtain the desired result of Fact 2 by substituting equations (48)–(51) into (47).
In view of (40), Facts 1 and 2 and Lemma 6 hold immediately.
The following theorem is derived from Lemmas 4–6.
Theorem 1. Let . Then,where
The explicit formulae of the spanning trees of are given below.
Theorem 2. Let . Then,
Proof . By Lemma 2, we have . Note thatHence, Theorem 2 immediately follows, along with Lemma 2.
4. Conclusion
In this study, we consider , 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 . New discoveries, developments, and advancements in research are still required. In the near future, we will be exploring a more complex chemistry network.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.