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), 2024. Published by Science Publishing Group |
Degree of a Vertex, Extended Adjacency Index, Tensor Product, Disjunction, Strong Product
| [1] | Adiga, C., Rakshith, B. R. (2018). Upper bounds for the extended energy of graphs and some extended equienergetic graphs. Opuscula Math 38 5-13. https://doi.org/10.7494/OpMath.2018.38.1.5 |
| [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 |
| [5] | Das, K. C., Xu, K., Cangul, I. N., Cevik, A. S., Graovac, A. (2013). On the Harary index of graph operations. Journal of inequalities and Applications 1 1-16. https://doi.org/10.1186/1029-242X-2013-339 |
| [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 |
| [8] | Eliasi, M., Taeri, B. (2009). Four new sums of graphs and their Wiener indices. Discrete Appl. Math. 157 794-803. https://doi.org/10.1016/j.dam.2008.07.001 |
| [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 |
| [10] | Gutman, I., Furtula, B., Das, K. C. (2017). Extended energy and its dependence on molecular structure. Can. J. Chem. 95 526-529. https://doi.org/10.1139/cjc-2016-0636 |
| [11] | Gutman, I. (2017). Relation between energy and extended energy of a graph. Int. J. Appl. Graph Theory 1 42-48. |
| [12] | Gutman, I. (2008). Lower bounds for Estrada index. Publ. Inst. Math. (Beograd) 83 1-7. DOI: 10.2298/PIM0897001G. |
| [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 |
| [14] | Hao, J. (2011). Theorems about Zagreb indices and modified Zagreb indices. MATCH Commun. Math. Comput. Chem. 65 659-670. |
| [15] | Liu, B., Huang, Y., Feng, J. (2012). A note on the randic spectral radius. MATCH Commun. Math. Comput. Chem. 68 913-916. |
| [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 |
| [19] | Tavakoli, M., Rahbarnia, F., Ashrafi, A. R. (2013). Note on strong product of graphs. Kragujevac Journal of mathematics 37 (1) 187-193. |
| [20] | Wang, Z., Mao, Y., Furtula, B., Wang, X. (2019). Bounds for the spectral radius and energy of extended adjacency matrix of graphs. Lin. Multilin. Algebra, DOI: 10. 1080/03081087.2019.1641464. https://doi.org/10.1080/03081087.2019.1641464 |
| [21] | Yang, Y. Q., Xu, L., Hu, C. Y. (1994). Extended adjacency matrix indices and their applications. J. Chem. Inf. Comput. Sci. 34 1140-1145. https://doi.org/10.1021/ci00021a020 |
| [22] | Yan, W., Yang, B., Yeh, Y. (2007). The behavior of Wiener indices and polynomials of graphs under five graph decorations. Appl. Math. Lett. 20 290-295. https://doi.org/10.1016/j.aml.2006.04.010 |
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
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
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
Share This Article
Twitter
Linked In
Facebook
Copy
Download@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}
}
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 -
Information
Email: