Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations

Chen Yang; Chongmin Li

doi:doi:10.11648/j.mcs.20251001.13

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Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations

Chen Yang^*, Chongmin Li

Published in Mathematics and Computer Science (Volume 10, Issue 1)

Received: 14 December 2024 Accepted: 25 December 2024 Published: 17 February 2025

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Abstract

The Wiener polarity index of a graph G, is the number of unordered pairs of vertices that are at distance 3 in G. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter d ≥ 3 are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.

Published in	Mathematics and Computer Science (Volume 10, Issue 1)
DOI	10.11648/j.mcs.20251001.13
Page(s)	19-25
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Wiener Polarity Index, Tree, Graph Entropy, Graph Operation

References

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[7]	Dehmer. M, Emmert-Streib. F, Shi. Y, Interrelations of graph distance measures based on topological indices. 9, 94985(2014). https://doi.org/10.1371/journal.pone.0094985
[8]	Dong. Yanni, Stijn. Cambie, On the main distance- based entropies: the eccentricity-entropy and Wiener- entropy. arxiv preprint arxiv. 2208, 12209(2022). https://doi.org/10.48550/arXiv.2208.12209
[9]	Hu. D, Li. X, Liu. X, Zhang. S, Extremality of graph entropy based on degrees of uniform hypergraphs with few edges. J. Acta Mathematica Sinica, English Series. 35(7), 1238-1250(2019). https://doi.org/CNKI:SUN:ACMS.0.2019-07-010
[10]	Pattabiraman. K, Nagarajan. S, Chendrasekharan. M, Zagreb indices and coindices of product graphs. J. Prime Res. Math. 10, 80-91 (2015). https://doi.org/10.1016/j.dam.2010.05.017
[11]	Dobrynin. A, Entringer. R, Gutman. I, Wiener index of trees: theory and application. Acta Appl Math. 66, 211- 249(2001). https://doi.org/10.1023/A:1010767517079
[12]	Swartz. E, Survey on the general Randi? index: extremal results and bounds. Rocky Mountain Journal of Mathematics. 52.4, 1177-1203 (2022). https://doi.org/10.1216/rmj.2022.52.1177
[13]	Wagner. S, Gutman. I, Maxima and minima of the hosoyaindexandthemerrifield-simmonsindex: Asurvey of results and techniques. Acta Appl Math. 112, 323- 346(2010). https://doi.org/10.1007/s10440-010-9575-5
[14]	Furtula. B, Gutman. I, Dehmer. M, On structure- sensitivity of degree-based topological indices. Applied Mathematics and Computation. 219, 8973-8978(2013). https://doi.org/10.1016/j.amc.2013.03.072
[15]	Dehmer. M, Mowshowitz. A, Emmert-Streib, F, Connections between classical and parametric network entropies. PLoS ONE 6, e15733 (2011). https://doi.org/10.1371/journal.pone.0015733

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APA Style

Yang, C., Li, C. (2025). Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Mathematics and Computer Science, 10(1), 19-25. https://doi.org/10.11648/j.mcs.20251001.13

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ACS Style

Yang, C.; Li, C. Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Math. Comput. Sci. 2025, 10(1), 19-25. doi: 10.11648/j.mcs.20251001.13

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AMA Style

Yang C, Li C. Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Math Comput Sci. 2025;10(1):19-25. doi: 10.11648/j.mcs.20251001.13

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@article{10.11648/j.mcs.20251001.13,
  author = {Chen Yang and Chongmin Li},
  title = {Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations},
  journal = {Mathematics and Computer Science},
  volume = {10},
  number = {1},
  pages = {19-25},
  doi = {10.11648/j.mcs.20251001.13},
  url = {https://doi.org/10.11648/j.mcs.20251001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20251001.13},
  abstract = {The Wiener polarity index of a graph G, is the number of unordered pairs of vertices that are at distance 3 in G. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter d ≥ 3 are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.},
 year = {2025}
}

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TY  - JOUR
T1  - Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations
AU  - Chen Yang
AU  - Chongmin Li
Y1  - 2025/02/17
PY  - 2025
N1  - https://doi.org/10.11648/j.mcs.20251001.13
DO  - 10.11648/j.mcs.20251001.13
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 19
EP  - 25
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20251001.13
AB  - The Wiener polarity index of a graph G, is the number of unordered pairs of vertices that are at distance 3 in G. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter d ≥ 3 are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.
VL  - 10
IS  - 1
ER  -

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Author Information

Chen Yang

School of Mathematics and Statistics, Qinghai Normal University, Xining, China

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Chongmin Li

School of Mathematics and Statistics, Qinghai Normal University, Xining, China

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Yang, C., Li, C. (2025). Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Mathematics and Computer Science, 10(1), 19-25. https://doi.org/10.11648/j.mcs.20251001.13

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ACS Style

Yang, C.; Li, C. Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Math. Comput. Sci. 2025, 10(1), 19-25. doi: 10.11648/j.mcs.20251001.13

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AMA Style

Yang C, Li C. Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations. Math Comput Sci. 2025;10(1):19-25. doi: 10.11648/j.mcs.20251001.13

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@article{10.11648/j.mcs.20251001.13,
  author = {Chen Yang and Chongmin Li},
  title = {Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations},
  journal = {Mathematics and Computer Science},
  volume = {10},
  number = {1},
  pages = {19-25},
  doi = {10.11648/j.mcs.20251001.13},
  url = {https://doi.org/10.11648/j.mcs.20251001.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20251001.13},
  abstract = {The Wiener polarity index of a graph G, is the number of unordered pairs of vertices that are at distance 3 in G. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter d ≥ 3 are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations
AU  - Chen Yang
AU  - Chongmin Li
Y1  - 2025/02/17
PY  - 2025
N1  - https://doi.org/10.11648/j.mcs.20251001.13
DO  - 10.11648/j.mcs.20251001.13
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 19
EP  - 25
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20251001.13
AB  - The Wiener polarity index of a graph G, is the number of unordered pairs of vertices that are at distance 3 in G. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter d ≥ 3 are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.
VL  - 10
IS  - 1
ER  -

Copy | Download