The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method

Tratnik, Niko

doi:10.1007/s10910-017-0749-5

Cited by 17 publications

(15 citation statements)

References 14 publications

(30 reference statements)

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“…, but the proof can be also found in Ref. .Lemma If u , v ∈ V(G) are two vertices , then .

d_{G} ((), u, v) = \sum_{i = 1}^{k} d_{G_{i}} ((),, ℓ_{i} ((), u), ℓ_{i} ((), v)) .

…”

Section: Preliminariesmentioning

confidence: 99%

Computing weighted Szeged and PI indices from quotient graphs

Tratnik

2019

Int J of Quantum Chemistry

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The weighted (edge‐)Szeged index and the weighted (vertex‐)PI index are modifications of the (edge‐)Szeged index and the (vertex‐)PI index, respectively, because they take into account also the vertex degrees. As the main result of this article, we prove that if G is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Θ*‐partition. If G is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system, the mentioned indices can be computed in sublinear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced.

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“…, but the proof can be also found in Ref. .Lemma If u , v ∈ V(G) are two vertices , then .

d_{G} ((), u, v) = \sum_{i = 1}^{k} d_{G_{i}} ((),, ℓ_{i} ((), u), ℓ_{i} ((), v)) .

…”

Section: Preliminariesmentioning

confidence: 99%

Computing weighted Szeged and PI indices from quotient graphs

Tratnik

2019

Int J of Quantum Chemistry

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“…This difference was computed in [9] for some families of polyhedral graphs. The Graovac-Pisanski index of nanostructures was studied in [1,2,15,16,17] and for some classes of fullerenes and fullerene-like molecules in [3,11,12]. In [13] the symmetry groups and Graovac-Pisanski index of some linear polymers were computed.…”

Section: Introductionmentioning

confidence: 99%

“…Upper and lower bounds for Graovac-Pisanski index were considered in [11]. In [7] and [16] Graovac-Pisanski index was further considered from computational point of view. Exact formulae for the Graovac-Pisanski index for some graph operations are presented in [4].…”

Section: Introductionmentioning

confidence: 99%

Unicyclic graphs with the maximal value of Graovac-Pisanski index

Knor¹,

Komorník²,

Škrekovski³

et al. 2019

Ars Math. Contemp.

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Let G be a graph and let Γ be its group of automorphisms. Graovac-Pisanski index of G is GP(G) = |V (G)| 2|Γ| u∈V (G) α∈Γ d(u, α(u)), where d(u, v) is the distance from u to v in G. One can observe that GP(G) = 0 if G has no nontrivial automorphisms, but it is not known which graphs attain the maximum value of Graovac-Pisanski index. In this paper we show that among unicyclic graphs on n vertices the n-cycle attains the maximum value of Graovac-Pisanski index.

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“…Consequently, the mentioned result can be used to develop very fast algorithms for calculating the edge-Wiener index of important chemical graphs or networks and also to easily find formulas in the closed form for some families of graphs. Such methods were recently developed also for other distance-based topological indices: the Wiener index [19], the revised (edge-)Szeged index [23], the degree distance [5], the Graovac-Pisanski index [25]. We should mention that instead of using our method for the edge-Wiener index one could use the method from [19] on the line graph.…”

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confidence: 99%

Generalized cut method for computing the edge-Wiener index

Tratnik¹

2020

Discrete Applied Mathematics

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The edge-Wiener index of a connected graph G is defined as the Wiener index of the line graph of G. In this paper it is shown that the edge-Wiener index of an edge-weighted graph can be computed in terms of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of weighted quotient graphs which are defined by a partition of the edge set that is coarser than Θ * -partition. Thus, already known analogous methods for computing the edge-Wiener index of benzenoid systems and phenylenes are greatly generalized. Moreover, reduction theorems are developed for the edge-Wiener index and the vertex-edge-Wiener index since they can be applied in order to compute a corresponding index of a (quotient) graph from the socalled reduced graph. Finally, the obtained results are used to find the closed formula for the edge-Wiener index of an infinite family of graphs.

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The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method

Cited by 17 publications

References 14 publications

Computing weighted Szeged and PI indices from quotient graphs

Computing weighted Szeged and PI indices from quotient graphs

Unicyclic graphs with the maximal value of Graovac-Pisanski index

Generalized cut method for computing the edge-Wiener index

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