Kirchhoff index, multiplicative degree‐Kirchhoff index and spanning trees of the linear crossed hexagonal chains

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).

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“…One may consult the recent work [22,23] and the references within. which implies that ℒ G = 1 k L G if G is k regular.…”

Section: K G ð þ= Pmentioning

confidence: 99%

Multiplicative degree‐Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC[2n, 2]

Sun

Wang

2019

Int J of Quantum Chemistry

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“…In this work, motivated by, we focus on two interesting types of molecular graphs: the penta‐graphene (penta‐C) R n and the pentagonal Möbius ring

R_{n}^{'}

(see Figure ). The penta‐graphene (penta‐C) R n is the graph obtained from the linear pentagonal chain L n by identifying the opposite lateral edges in an ordered way, that is, penta‐graphene (penta‐C) R n is obtained from L n by identifying the vertex 1 and (2 n + 1), vertex 1 ′ and (2 n + 1) ′ , respectively.…”

Section: Introductionmentioning

confidence: 99%

Study on the normalized Laplacian of a penta‐graphene with applications

Zaman

Sun

et al. 2020

Int J of Quantum Chemistry

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Let L n denote a linear pentagonal chain with 2n pentagons. The penta‐graphene (penta‐C), denoted by R n is the graph obtained from L n by identifying the opposite lateral edges in an ordered way, whereas the pentagonal Möbius ring Rn′ is the graph obtained from the L n by identifying the opposite lateral edges in a reversed way. In this paper, through the decomposition theorem of the normalized Laplacian characteristic polynomial and the relationship between its roots and the coefficients, an explicit closed‐form formula of the multiplicative degree‐Kirchhoff index (resp. Kemeny's constant, the number of spanning trees) of R n is obtained. Furthermore, it is interesting to see that the multiplicative degree‐Kirchhoff index of R n is approximately 13 of its Gutman index. Based on our obtained results, all the corresponding results are obtained for Rn′.

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“…Hence, it is interesting to obtain the closed‐form formula for the multiplicative degree‐Kirchhoff index of graph G . For recent advances on this topic, one may be referred to and the references with in.…”

Section: Introductionmentioning

confidence: 99%

“…[8] Clearly, it is much more difficult to use algorithms to compute the multiplicative degree-Kirchhoff index in a graph. Hence, it is interesting to obtain the closedform formula for the multiplicative degree-Kirchhoff index of graph G. For recent advances on this topic, one may be referred to [17][18][19][20][21][22][23][24][25][26] and the references with in.…”

Section: Introductionmentioning

confidence: 99%

On the normalized Laplacian of Möbius phenylene chain and its applications

Lei

Geng

et al. 2019

Int J of Quantum Chemistry

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Let Ln6,4 denote a molecular graph of linear [n] phenylene with n hexagons and n squares, and let the Möbius phenylene chain H0emMn6,4 be the graph obtained from the Ln6,4 by identifying the opposite lateral edges in reversed way. Utilizing the decomposition theorem of the normalized Laplacian characteristic polynomial, we study the normalized Laplacian spectrum of H0emMn6,4, which consists of the eigenvalues of two symmetric matrices ℒ R and ℒ Q of order 3n. By investigating the relationship between the roots and coefficients of the characteristic polynomials of the two matrices above, we obtain an explicit closed‐form formula of the multiplicative degree‐Kirchhoff index as well as the number of spanning trees of H0emMn6,4. Furthermore, we determine the limited value for the quotient of the multiplicative degree‐Kirchhoff index and the Gutman index of H0emMn6,4.

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Kirchhoff index, multiplicative degree‐Kirchhoff index and spanning trees of the linear crossed hexagonal chains

Cited by 45 publications

References 35 publications

Multiplicative degree‐Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC[2n, 2]

Multiplicative degree‐Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC[2n, 2]

Study on the normalized Laplacian of a penta‐graphene with applications

On the normalized Laplacian of Möbius phenylene chain and its applications

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