Tutte polynomials of alternating polycyclic chains

Chen, Hanlin; Guo, Qiuzhi

doi:10.1007/s10910-019-01069-2

Cited by 13 publications

(15 citation statements)

References 24 publications

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“…In the following, we use

{PH}_{n}

to denote a phenylene chain with

n

hexagons and

n - 1

squares, that is,

{PH}_{n} \in G_{n}^{6, 4}

. Thus, by Theorem 6, we have the following result [10]. Corollary Let

{PH}_{n}

be a phenylene chain with

n

hexagons and

n - 1

squares, then the Tutte polynomial of

{PH}_{n}

T ((), {PH}_{n}; x, y) goodbreak= \frac{2 ω - γ ((), α goodbreak- \sqrt{Δ})}{normalΔ + α \sqrt{Δ}} {(\frac{α + \sqrt{Δ}}{2})}^{n} goodbreak+ \frac{2 ω - γ ((), α goodbreak+ \sqrt{Δ})}{normalΔ - α \sqrt{Δ}} {(\frac{α - \sqrt{Δ}}{2})}^{n},

where

normalΔ = α^{2} + 4 β

α = x^{6} + 2 x^{5} + 3 x^{4} + x^{3} y + 3 x^{3} + x^{2} y + 3 x^{2} + 2 italicxy + 2 x + y^{2} + 2 y + 1, β = - x^{6} y^{2}, γ = y + x + x^{2} + x^{3} + x^{4} + x^{5},

and

…”

Section: The Tutte Polynomial Of Alternating Polycyclic Chainsmentioning

confidence: 99%

“…Let

t ((), G) = T ((), G; 1, 1)

be the number of spanning trees of a graph G . Then by Corollary 8 we get the number of spanning trees of

{PH}_{n}

.Corollary See Reference [10]. Let

{PH}_{n}

be a phenylene chain with n hexagons and

n - 1

squares, then the number of spanning trees of

{PH}_{n}

t ((), {italicPH}_{n}) goodbreak= \frac{((), 33 goodbreak+ 6 \sqrt{30}) {(11 + 2 \sqrt{30})}^{n}}{120 + 22 \sqrt{30}} goodbreak+ \frac{((), 33 goodbreak- 6 \sqrt{30}) {(11 - 2 \sqrt{30})}^{n}}{120 - 22 \sqrt{30}} .

…”

Section: The Tutte Polynomial Of Alternating Polycyclic Chainsmentioning

confidence: 99%

“…Dobrynin and Vesnin [4] given a general calculating scheme of polynomials by deleting and contracting edges for uniform polycyclic chains. Chen and Guo [10] derived the Tutte polynomials of certain alternating polycyclic chains by using a direct deletion‐contraction‐based way. In Reference [11], the author presented a splitting formula for the Tutte polynomial of a class of special compound graphs, and as an application the explicit analytical expressions for the Tutte polynomials of catacondensed benzenoid systems with given number branching hexagons are also determined.…”

Section: Introductionmentioning

confidence: 99%

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The Tutte polynomial of phenylene systems with given number of branching hexagons

Chen

2022

Int J of Quantum Chemistry

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Polynomial graph invariants have been confirmed to have important applications in quantum chemistry and biological information. One of the famous polynomial graph invariants is the Tutte polynomial which gives abundant graph‐theoretical information of the underlying graph. In this paper, we first give a simpler and more efficient method to get the Tutte polynomials of alternating polycyclic chains. Then we obtain the explicit expressions for the Tutte polynomials and the number of spanning trees of phenylene systems with given number of branching hexagons. Moreover, we determine the extremal values of the number of spanning trees among the phenylene systems with given one branching hexagon and two branching hexagons. The corresponding extremal phenylene systems are characterized, respectively.

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“…In the following, we use

{PH}_{n}

to denote a phenylene chain with

n

hexagons and

n - 1

squares, that is,

{PH}_{n} \in G_{n}^{6, 4}

. Thus, by Theorem 6, we have the following result [10]. Corollary Let

{PH}_{n}

be a phenylene chain with

n

hexagons and

n - 1

squares, then the Tutte polynomial of

{PH}_{n}

T ((), {PH}_{n}; x, y) goodbreak= \frac{2 ω - γ ((), α goodbreak- \sqrt{Δ})}{normalΔ + α \sqrt{Δ}} {(\frac{α + \sqrt{Δ}}{2})}^{n} goodbreak+ \frac{2 ω - γ ((), α goodbreak+ \sqrt{Δ})}{normalΔ - α \sqrt{Δ}} {(\frac{α - \sqrt{Δ}}{2})}^{n},

where

normalΔ = α^{2} + 4 β

α = x^{6} + 2 x^{5} + 3 x^{4} + x^{3} y + 3 x^{3} + x^{2} y + 3 x^{2} + 2 italicxy + 2 x + y^{2} + 2 y + 1, β = - x^{6} y^{2}, γ = y + x + x^{2} + x^{3} + x^{4} + x^{5},

and

…”

Section: The Tutte Polynomial Of Alternating Polycyclic Chainsmentioning

confidence: 99%

“…Let

t ((), G) = T ((), G; 1, 1)

be the number of spanning trees of a graph G . Then by Corollary 8 we get the number of spanning trees of

{PH}_{n}

.Corollary See Reference [10]. Let

{PH}_{n}

be a phenylene chain with n hexagons and

n - 1

squares, then the number of spanning trees of

{PH}_{n}

t ((), {italicPH}_{n}) goodbreak= \frac{((), 33 goodbreak+ 6 \sqrt{30}) {(11 + 2 \sqrt{30})}^{n}}{120 + 22 \sqrt{30}} goodbreak+ \frac{((), 33 goodbreak- 6 \sqrt{30}) {(11 - 2 \sqrt{30})}^{n}}{120 - 22 \sqrt{30}} .

…”

Section: The Tutte Polynomial Of Alternating Polycyclic Chainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Tutte polynomial of phenylene systems with given number of branching hexagons

Chen

2022

Int J of Quantum Chemistry

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“…In [10], an explicit expression for the Tutte polynomial of catacondensed benzenoid systems with exactly one branched hexagon was obtained. In [4], Tutte polynomials of alternating polycyclic chains were obtained. Recently, a reduction formula for Tutte polynomial of any catacondensed benzenoid system was obtained by three classes of transfer matrices in [18].…”

Section: Introductionmentioning

confidence: 99%

“…3 x+ 3y 3 + 3y 2 x 7 + 9y 2 x 6 + 15y 2 x 5 + 20y 2 x 4 + 21y 2 x 3 +15y 2 x 2 +9y 2 x+3y 2 +2yx 11 +8yx 10 +18yx 9 +32yx 8 +46yx 7 +53yx 6 + 52yx 5 + 43yx 4 + 28yx 3 + 15yx 2 + 6yx + y + x 15 + 4x 14 + 10x 13 + 20x 12 + 33x 11 + 46x 10 + 56x 9 + 60x 8 + 56x 7 + 46x 6 + 33x 5 + 20x 4 + 10x 3 + 4x 2 + x, J = x 4 (y 3 + y 2 x 4 + 3y 2 x 3 + 3y 2 x 2 + 3y 2 x + 2y 2 + yx 8 + 4yx 7 + 7yx 6 + 10yx 5 + 12yx 4 + 10yx 3 + 7yx 2 + 4yx + y + x 12 + 3x 11 + 6x 10 + 10x 9 + 14x 8 + 16x 7 + 16x 6 + 14x 5 + 10x4 …”

mentioning

confidence: 99%

Tutte polynomials of fan-like graphs with applications in benzenoid systems

Zhang

2021

Preprint

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The cut method on hypergraphs for the Wiener index

Klavžar

Romih

2023

J Math Chem

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The cut method has been proved to be extremely useful in chemical graph theory. In this paper the cut method is extended to hypergraphs. More precisely, the method is developed for the Wiener index of k-uniform partial cube-hypergraphs. The method is applied to cube-hypergraphs and hypertrees. Extensions of the method to hypergraphs arising in chemistry which are not necessary k-uniform and/or not necessary linear are also developed.

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Tutte polynomials of alternating polycyclic chains

Cited by 13 publications

References 24 publications

The Tutte polynomial of phenylene systems with given number of branching hexagons

The Tutte polynomial of phenylene systems with given number of branching hexagons

Tutte polynomials of fan-like graphs with applications in benzenoid systems

The cut method on hypergraphs for the Wiener index

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