The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains

Huang, Jing; Li, Shuchao; Li, Xuechao

doi:10.1016/j.amc.2016.05.024

Cited by 56 publications

(34 citation statements)

References 24 publications

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“…There is extensive literature available on works related to Kf (G), Kf * (G) and Kf + (G), one may be referred to [19,20,21,29,31] for more detailed information. For a graph G, define the random walks on G as the Markov chain X k , k ≥ 0, that from its current vertex x jumps to its adjacent vertex with probability 1/d(x).…”

Section: 2)mentioning

confidence: 99%

Further results on the expected hitting time, the cover cost and the related invariants of graphs

Huang

Xie

2019

Discrete Mathematics

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A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an n-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified, respectively.

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Section: 2)mentioning

confidence: 99%

Further results on the expected hitting time, the cover cost and the related invariants of graphs

Huang

Xie

2019

Discrete Mathematics

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“…Let L ( G ), L A ( G ) , and L S ( G ) be defined as above. Then,

ψ (();, G, x) = normalΦ ((), L_{A} ((), G)) \cdot normalΦ ((), L_{S} ((), G)) .

Lemma . Let ℒ( G ), ℒ A ( G ) , and ℒ S ( G ) be defined as above.…”

Section: Laplacian Polynomial Decomposition and Normalized Laplacian mentioning

confidence: 99%

“…In Refs. , explicit closed formulas for the Kirchhoff index, multiplicative degree‐Kirchhoff index and the number of spanning trees of linear polyomino chains, linear pentagonal chains, and linear hexagonal chains are determined, respectively. Interesting enough, the Kirchhoff (resp.…”

Section: Introductionmentioning

confidence: 99%

“…In this article, inspired by the works in Refs. [7, 23, 24, 26, and 28–31], we obtain the decomposition theorems for the Laplacian polynomial and normalized Laplacian polynomial of H n . Based on these results, explicit formulas for the Kirchhoff index, multiplicative degree‐Kirchhoff index and the number of spanning trees of H n are determined, respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Pan

2018

Int J of Quantum Chemistry

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Let H n be a linear crossed hexagonal chain with n crossed hexagonals. In this article, we find that the Laplacian (resp. normalized Laplacian) spectrum of H n consists of the eigenvalues of a symmetric tridiagonal matrix of order 2n + 1 and a diagonal matrix of order 2n + 1. Based on the properties of these matrices, significant closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of H n are derived. Finally, we show that the Kirchhoff (resp. multiplicative degree-Kirchhoff ) index of H n is approximately one quarter of its Wiener (resp. Gutman) index.

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“…Some mathematical properties and extremal problems on ξ R (G) are considered. Some interested properties and relations among these Kirchhoffian indices are obtained, see [8,9,13,14] and references therein. Motivated by these works above, we defined a new index ξ * R (G) [7] from (1.1) by taking f G (x, y) = ε(x) • ε(y), name it as multiplicative eccentricity resistance-distance, and some mathematical properties on ξ * R (G) were studied, as an application, the extremal graphs with minimum and second minimum ξ * R (G)-value in Cat(n; t) were characterized.…”

Section: Introductionmentioning

confidence: 99%

The first two cacti with larger multiplicative eccentricity resistance-distance

Hong

Zhu

2019

Filomat

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For a connected graph G, the multiplicative eccentricity resistance-distance ξ * R (G) is defined as ξ * R (G) = {x,y}⊆V(G) ε(x) • ε(y)R G (x, y), where ε(•) is the eccentricity of the corresponding vertex and R G (x, y) is the effective resistance between vertices x and y. A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Let Cat(n; t) be the set of cacti possessing n vertices and t cycles, where 0 ≤ t ≤ n−1 2. In this paper, we first introduce some edge-grafting transformations which will increase ξ * R (G). As their applications, the extremal graphs with maximum and second-maximum ξ * R (G)-value in Cat(n; t) are characterized, respectively.

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The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains

Cited by 56 publications

References 24 publications

Further results on the expected hitting time, the cover cost and the related invariants of graphs

Further results on the expected hitting time, the cover cost and the related invariants of graphs

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

The first two cacti with larger multiplicative eccentricity resistance-distance

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