On normalized Laplacians, multiplicative degree‐Kirchhoff indices, and spanning trees of the linear [<i>n</i>]phenylenes and their dicyclobutadieno derivatives

Int J of Quantum Chemistry

Wang

2019

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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).

“…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%

“…This novel graph matrix has attracted much attention. One may consult the recent work and the references within.…”

Section: Introductionmentioning

confidence: 99%

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

Int J of Quantum Chemistry

Wang

2019

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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

Int J of Quantum Chemistry

et al. 2020

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′.

“…This novel graph matrix has attracted much attention. One may consult the recent work [18,19] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On split graphs with three or four distinct (normalized) Laplacian eigenvalues

J of Combinatorial Designs

2020

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It is well known to us that a graph of diameter l has at least l + 1 eigenvalues. A graph is said to be Laplacian (resp, normalized Laplacian) l‐extremal if it is of diameter l having exactly l + 1 distinct Laplacian (resp, normalized Laplacian) eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Each split graph is of diameter at most 3. In this paper, we completely classify the connected bidegreed Laplacian (resp, normalized Laplacian) 2‐extremal (resp, 3‐extremal) split graphs using the association of split graphs with combinatorial designs. Furthermore, we identify connected bidegreed split graphs of diameter 2 having just four Laplacian (resp, normalized Laplacian) eigenvalues.