Let L n denote the linear hexagonal chain containing n hexagons. Then identifying the opposite lateral edges of L n in ordered way yields TUHC[2n, 2], the zigzag polyhex nanotube, whereas identifying those of L n in reversed way yields M n , the hexagonal Möbius chain. In this article, we first obtain the explicit formulae of the multiplicative degree-Kirchhoff index, the Kemeny's invariant, the total number of spanning trees of TUHC[2n, 2], respectively. Then we show that the multiplicative degree-Kirchhoff index of TUHC[2n, 2] is approximately one-third of its Gutman index. Based on these obtained results we can at last get the corresponding results for M n . K E Y W O R D S multiplicative degree-Kirchhoff index, normalized Laplacian, spanning tree, zigzag polyhex nanotube AMS Classification 05C35, 05C12
| INTRODUCTIONChemical structures are commodiously represented by graphs, where atoms correlate with vertices and chemical bonds correlate with edges. This manifestation inherits much valuable information on chemical properties of molecules. In quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies one may see that many chemical and physical qualities of molecules are well corresponding to graph theoretical parameters that are termed topological indices. One of such graph theoretical parameters is the multiplicative degree-Kirchhoff index (see Ref. [1]). On the other hand, the enumeration of spanning trees of a graph is a very important problem in statistical physics. [2] It is nice to see that the number of spanning trees of a graph is closely related to the multiplicative degree-Kirchhoff index. The normalized Laplacian is the bridge connecting them.In this article, G is a simple, finite, and connected graph, with vertex set V = V G and edge set E = E G . The order |V| of G is denoted by n = |V| and the size |E| of G is written as ε = |E|. Unless otherwise stated, one may follow the terminologies and notations in Refs. [3][4][5].Given a graph G, its adjacency matrix A G is a 0-1 square matrix of order n whose (k, l)-entry is 1 if and only if kl 2 E. Let D G = diag(d 1 , d 2 , … , d n ) be the diagonal matrix whose diagonal entries are the degrees in G. Then the matrix L G = D G − A G is called the Laplacian matrix of G, whose eigenvalues are ordered as μ 1 ≤ μ 2 ≤ Á Á Á ≤ μ n . It is well known that μ 1 = 0. In particular, μ 2 > 0 if and only if G is connected. One may consult the recent work [6,7] and the references within for more developments on L G .Let M be a matrix of order m × n. Then we write M(X|Y) to denote the submatrix of M obtained by deleting the rows and columns in X and Y, respectively, where X (resp. Y) is a subset of {1, 2, … , m} (resp. {1, 2, … , n}). In particular, if X = {r}, Y = {t}, then let M(X|Y) = M(r|t).
