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The Extended Adjacency Indices for Several Types of Graph Operations

Received: 4 January 2023     Accepted: 29 January 2023     Published: 14 February 2023
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Abstract

Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index EA(G) is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.

Published in International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 1)
DOI 10.11648/j.ijssam.20230801.12
Page(s) 7-11
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), 2023. Published by Science Publishing Group

Keywords

Degree of a Vertex, Extended Adjacency Index, Tensor Product, Disjunction, Strong Product

References
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[2] Bamdad, H., Ashraf, F., Gutman, I. (2010). Lower bounds for Estrada index and Laplacian Estrada index. Applied Mathematics Letters 23 739-742. https://doi.org/10.1016/j.aml.2010.01.025
[3] Bondy, J. A., Murty, U. S. R. (2008). Graph Theory. London, Spring.
[4] Das, K. C., Gutman, I., Furtula, B. (2017). On spectral radius and energy of extended adjacency matrix of graphs. Appl. Math. Comput. 296 116-123. https://doi.org/10.1016/j.aml.2010.01.025
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[6] Das, K. C., Xu, K., Nam, J. (2015). Zagreb indices of graphs. Front. Math. China 10 567-582. https://doi.org/10.1007/s11464-015-0431-9
[7] Dobrynin, A. A., Entringer, R., Gutman, I. (2001). Wiener index of trees. Theory and applications, Acta Appl. Math. 66 211-249. https://doi.org/10.1023/A:1010767517079
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[9] Furtula, B., Gutman, I. (2015). A forgotten topological index. J. Math. Chem. 53 1184-1190. https://doi.org/10.1007/s10910-015-0480-z
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[13] Gupta, C. K., Lokesha, V., Shetty, S. B. (2016). On the graph of nilpotent matrix group of length one. Discrete Mathematics, Algorithms and applications 8 (1) 1-13. https://doi.org/10.1142/S1793830916500099
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[16] Li, X., Li, Y. (2013). The asympotic behavior of Estrada index for trees. Bull. Malays. Math. Sci. Soc. (2) 36 97-106.
[17] Li, X., Shi, Y. (2008). A survey on the Randic index. MATCH Commun. Math. Comput. Chem. 59 127-156.
[18] Pattabiraman, K., Paulraja, P. (2012). On some topological indices of the tensor products of graphs. Discrete Applied Mathematics 160 267-279. https://doi.org/10.1016/j.dam.2011.10.020
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  • APA Style

    Feng Fu, Bo Deng, Hongyu Zhang. (2023). The Extended Adjacency Indices for Several Types of Graph Operations. International Journal of Systems Science and Applied Mathematics, 8(1), 7-11. https://doi.org/10.11648/j.ijssam.20230801.12

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    ACS Style

    Feng Fu; Bo Deng; Hongyu Zhang. The Extended Adjacency Indices for Several Types of Graph Operations. Int. J. Syst. Sci. Appl. Math. 2023, 8(1), 7-11. doi: 10.11648/j.ijssam.20230801.12

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    AMA Style

    Feng Fu, Bo Deng, Hongyu Zhang. The Extended Adjacency Indices for Several Types of Graph Operations. Int J Syst Sci Appl Math. 2023;8(1):7-11. doi: 10.11648/j.ijssam.20230801.12

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  • @article{10.11648/j.ijssam.20230801.12,
      author = {Feng Fu and Bo Deng and Hongyu Zhang},
      title = {The Extended Adjacency Indices for Several Types of Graph Operations},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {8},
      number = {1},
      pages = {7-11},
      doi = {10.11648/j.ijssam.20230801.12},
      url = {https://doi.org/10.11648/j.ijssam.20230801.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230801.12},
      abstract = {Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index EA(G) is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - The Extended Adjacency Indices for Several Types of Graph Operations
    AU  - Feng Fu
    AU  - Bo Deng
    AU  - Hongyu Zhang
    Y1  - 2023/02/14
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ijssam.20230801.12
    DO  - 10.11648/j.ijssam.20230801.12
    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
    SP  - 7
    EP  - 11
    PB  - Science Publishing Group
    SN  - 2575-5803
    UR  - https://doi.org/10.11648/j.ijssam.20230801.12
    AB  - Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index EA(G) is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.
    VL  - 8
    IS  - 1
    ER  - 

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Author Information
  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

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

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

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