Abstract
Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .
Highlights
With the discovery of carbon nanotubes, there has been a practical interest in the study of concerning structures based on hexagon networks
Peng et al [3] had studied the Kirchhoff index and related spanning trees for the linear phenylenes which are constructed from polyominoes and hexagons
Tips of carbon nanotubes are closed by pentagons
Summary
With the discovery of carbon nanotubes, there has been a practical interest in the study of concerning structures based on hexagon networks. 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 [11,12,13,14,15,16,17,18,19,20]. At this point, we slightly observe some properties of the Kirchhoff index. One finds that Kirchhoff index of a graph G is closely related to the Laplacian eigenvalue. We obtain the number of spanning trees of Hn
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