The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains

Huang, Jing; Li, Shuchao; Sun, Liqun

doi:10.1016/j.dam.2016.02.019

Cited by 62 publications

(44 citation statements)

References 24 publications

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“…Huang and Li obtained the following decomposition theorem of the normalized Laplacian characteristic polynomial. This is an exact analogue of the decomposition theorem of the Laplacian characteristic polynomial …”

Section: Normalized Laplacian Characteristic Polynomial and Some Prelmentioning

confidence: 99%

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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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Section: Normalized Laplacian Characteristic Polynomial and Some Prelmentioning

confidence: 99%

“…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%

On the normalized Laplacian of Möbius phenylene chain and its applications

Lei

Geng

et al. 2019

Int J of Quantum Chemistry

Self Cite

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“…This is an exact analogue of the decomposition theorem of the Laplacian characteristic polynomial Theorem Let G be a graph, and let ℒ( G ), ℒ A ( G ), ℒ S ( G ) be defined as above. Then,

Ψ (G; x) = Φ (ℒ_{A} (G)) \cdot Φ (ℒ_{S} (G)) .

Lemma Let G be an n‐vertex connected graph with m edges, and let 0 = λ 1 < λ 2 ≤ ⋯ ≤ λ n be the eigenvalues of ℒ(G) .…”

Section: Normalized Laplacian Characteristic Polynomial and Some Prelmentioning

confidence: 99%

“…Recently, some closed‐form formulas for the multiplicative degree‐Kirchhoff index have been obtained for some types of graphs. One may be referred to References for detailed information.…”

Section: Introductionmentioning

confidence: 99%

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

Wei

Shi-qun

2018

Int J of Quantum Chemistry

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Let Ln6,6 be the molecular graph of the linear [n]phenylene with n hexagons and n − 1 squares, and let Ln4,4 be the graph obtained by attaching four‐membered rings to the terminal hexagons of Ln6,6. In this article, the normalized Laplacian spectrum of Ln6,6 consisting of the eigenvalues of two symmetric tridiagonal matrices of order 3n is determined. An explicit closed‐form formula of the multiplicative degree‐Kirchhoff index (respectively the number of spanning trees) of Ln6,6 is derived. Similarly, explicit closed‐form formulas of the multiplicative degree‐Kirchhoff index and the number of spanning trees of Ln4,4 are obtained. It is interesting to see that the multiplicative degree‐Kirchhoff index of Ln6,6 (respectively Ln4,4) is approximately to one half of its Gutman index.

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“…(For an example of the very latest work on spanning trees and their relations to graph eigenvalues, please see Refs. [51,52] and the articles cited in them.) Let D be the (v × v) degree matrix [27,53,54] of some labelled graph G (with v vertices and e edges), and let A be the (v × v) vertex-adjacency matrix of G, so labelled.…”

Section: Introductionmentioning

confidence: 99%

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

Kirby¹,

Mallion²,

Pollak³

et al. 2016

Croat. Chem. Acta

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What Kirchhoff actually did concerning spanning trees in the course of his classic paper in the 1847 Annalen der Physik und Chemie has, to some extent, long been shrouded in myth in the literature of Graph Theory and Mathematical Chemistry. In this review, Kirchhoff's manipulation of the equations that arise from application of his two celebrated Laws of electrical circuits -formulated in the middle of the 19 th century -is related to 20 th -and 21 st -century work on the enumeration of spanning trees. It is shown that matrices encountered in an analysis of what Kirchhoff really did include (a) the Kirchhoff (Laplacian, Admittance) matrix, K, that features in the well-known Matrix Tree Theorem, (b) the matrix G encountered in the theorem of Gutman, Mallion & Essam (1983), applicable only to planar graphs, and (c) the analogous matrix M that arises in the Cycle Theorem (Kirby et al. 2004), a theorem that applies to graphs of any genus. It is concluded that Kirchhoff himself was not interested in counting spanning trees, and, accordingly, he did not explicitly do so. Nevertheless, it is shown how the modulus of the determinant of a certain matrix (here denoted by the label C') -associated with the linear equations arising from application of Kirchhoff's two Laws -is numerically equal to the number of spanning trees in the graph representing the connectivity of the electrical network being studied. Kirchhoff did, however, invoke the concept of spanning trees, introducing them in a complementary fashion by referring to the chords that must be removed from the original graph in order to form such trees. It is further emphasised that, in choosing the cycles in the network being studied, around which to apply his circuit Law, Kirchhoff explicitly selected what would now be called a 'Fundamental System of Cycles'.

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The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains

Cited by 62 publications

References 24 publications

On the normalized Laplacian of Möbius phenylene chain and its applications

On the normalized Laplacian of Möbius phenylene chain and its applications

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

What Kirchhoff Actually did Concerning Spanning Trees in Electrical Networks and its Relationship to Modern Graph-Theoretical Work

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