The weighted vertex PI index of bicyclic graphs

Gang, Ma; Bian, Qiuju; Wang, Jianfeng

doi:10.1016/j.dam.2018.03.087

Cited by 5 publications

(4 citation statements)

References 21 publications

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“… and Refs. , respectively. However, it seems natural to consider also the weighted edge‐Szeged index and the weighted PI index :

\begin{matrix} {right left}true \\ {italicwSz}_{e} (G) = \underset{e = uv \in E (G)}{false\sum} (\deg (u) + \deg (v)) m_{u} (e) m_{v} (e), \\ wPI (G) = \underset{e = uv \in E (G)}{false\sum} (\deg (u) + \deg (v)) (m_{u} (e) + m_{v} (e)) . \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

“…For some research on these indices see [3,25,27,28] and [23,24,26,34], respectively. However, it seems natural to consider also the weighted edge-Szeged index and the weighted PI index :…”

Section: Introductionmentioning

confidence: 99%

“…[14][15][16][17] and Refs. [18][19][20][21], respectively. However, it seems natural to consider also the weighted edge-Szeged index and the weighted PI index:…”

mentioning

confidence: 99%

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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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“… and Refs. , respectively. However, it seems natural to consider also the weighted edge‐Szeged index and the weighted PI index :

\begin{matrix} {right left}true \\ {italicwSz}_{e} (G) = \underset{e = uv \in E (G)}{false\sum} (\deg (u) + \deg (v)) m_{u} (e) m_{v} (e), \\ wPI (G) = \underset{e = uv \in E (G)}{false\sum} (\deg (u) + \deg (v)) (m_{u} (e) + m_{v} (e)) . \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

“…For some research on these indices see [3,25,27,28] and [23,24,26,34], respectively. However, it seems natural to consider also the weighted edge-Szeged index and the weighted PI index :…”

Section: Introductionmentioning

confidence: 99%

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Computing weighted Szeged and PI indices from quotient graphs

Tratnik

2019

Int J of Quantum Chemistry

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“…In the same paper, the exact expressions for the weighted vertex PI index of the Cartesian product of graphs are also given. In [43] and [28], bounds and the extremal graphs on the weighted vertex PI index of connected unicyclic and bicyclic graph are given respectively. In [34], the exact formula for the weighted vertex PI index of corona product of two connected graphs is obtained.…”

Section: Introductionmentioning

confidence: 99%

Bounds on the weighted vertex PI index of cacti graphs

Gang

Bian

Wang

2019

Filomat

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The weighted vertex PI index of a graph G is defined by PI w (G) = e=uv∈E(G) (d G (u) + d G (v))(n u (e|G) + n v (e|G)) where d G (u) denotes the vertex degree of u and n u (e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

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