On the Edge Connectivity of Semi-strong Product Graphs

Qiaoling Wang; Haizhen Ren

doi:doi:10.11648/j.acm.20251406.15

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On the Edge Connectivity of Semi-strong Product Graphs

Qiaoling Wang

, Haizhen Ren^*

Published in Applied and Computational Mathematics (Volume 14, Issue 6)

Received: 9 September 2025 Accepted: 23 October 2025 Published: 11 December 2025

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Abstract

The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs and , their semi-strong product is a graph whose vertex set is , and the edge set is defined as follows: if and are two vertices in the semi-strong product , then there is an edge between them if and only if and and are adjacent in , or and are adjacent in and and are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.

Published in	Applied and Computational Mathematics (Volume 14, Issue 6)
DOI	10.11648/j.acm.20251406.15
Page(s)	356-359
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

Edge Connectivity, Graph Product, Semi-strong Product

References

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[4]	Anastasiia, K., Bonaventura, D. M., Steffen, Z. and Volker, M. Fault Tolerance Placement in the Internet of Things, Proc. ACM Manag. 2024, 138, 1-29. HYPERLINK " https://doi.org/10.1145/365494" https://doi.org/10.1145/3654941
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[7]	Gao, Z., Jiang, C. and Zhang, J, et al. Hierarchical graph learning for protein–protein interaction, Nat Commun. 2023, 14, 1093. HYPERLINK " https://doi.org/10.1038/s41467-023-36736-1" https://doi.org/10.1038/s41467-023-36736-1
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[11]	Yang, C. and Xu, J. M. Connectivity and edge-connectivity of strong product graphs, J. Univ. Sci. Technol. China. 2008, 38(5), 449-455. http://staff.ustc.edu.cn/~xujm/200804.pdf
[12]	Bresar, B. and Spacapan, S. Edge-connectivity of strong products of graphs, Discus siones Mathematicae Graph Theory. 2007, 27(2): 333-343. HYPERLINK " https://doi.org/10.7151/dmgt.1365" https://doi.org/10.7151/dmgt.1365
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Wang, Q., Ren, H. (2025). On the Edge Connectivity of Semi-strong Product Graphs. Applied and Computational Mathematics, 14(6), 356-359. https://doi.org/10.11648/j.acm.20251406.15

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Wang, Q.; Ren, H. On the Edge Connectivity of Semi-strong Product Graphs. Appl. Comput. Math. 2025, 14(6), 356-359. doi: 10.11648/j.acm.20251406.15

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AMA Style

Wang Q, Ren H. On the Edge Connectivity of Semi-strong Product Graphs. Appl Comput Math. 2025;14(6):356-359. doi: 10.11648/j.acm.20251406.15

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@article{10.11648/j.acm.20251406.15,
  author = {Qiaoling Wang and Haizhen Ren},
  title = {On the Edge Connectivity of Semi-strong Product Graphs},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {6},
  pages = {356-359},
  doi = {10.11648/j.acm.20251406.15},
  url = {https://doi.org/10.11648/j.acm.20251406.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251406.15},
  abstract = {The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs  and , their semi-strong product  is a graph whose vertex set is , and the edge set  is defined as follows: if  and  are two vertices in the semi-strong product , then there is an edge between them if and only if  and  and  are adjacent in , or  and  are adjacent in  and  and  are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.},
 year = {2025}
}

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TY  - JOUR
T1  - On the Edge Connectivity of Semi-strong Product Graphs
AU  - Qiaoling Wang
AU  - Haizhen Ren
Y1  - 2025/12/11
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251406.15
DO  - 10.11648/j.acm.20251406.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 356
EP  - 359
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251406.15
AB  - The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs  and , their semi-strong product  is a graph whose vertex set is , and the edge set  is defined as follows: if  and  are two vertices in the semi-strong product , then there is an edge between them if and only if  and  and  are adjacent in , or  and  are adjacent in  and  and  are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.
VL  - 14
IS  - 6
ER  -

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Author Information

Qiaoling Wang

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

Contact Email

http://orcid.org/0009-0001-6058-7857
Haizhen Ren

School of Mathematics and Statistics, Qinghai Normal University, Xining, China;The State Key Laboratory of Tibetan Intelligence, Xining, China;Academy of Plateau, Science and Sustainability, Xining, China

Contact Email

http://orcid.org/0000-0001-5609-5924

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APA Style

Wang, Q., Ren, H. (2025). On the Edge Connectivity of Semi-strong Product Graphs. Applied and Computational Mathematics, 14(6), 356-359. https://doi.org/10.11648/j.acm.20251406.15

Copy | Download

ACS Style

Wang, Q.; Ren, H. On the Edge Connectivity of Semi-strong Product Graphs. Appl. Comput. Math. 2025, 14(6), 356-359. doi: 10.11648/j.acm.20251406.15

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AMA Style

Wang Q, Ren H. On the Edge Connectivity of Semi-strong Product Graphs. Appl Comput Math. 2025;14(6):356-359. doi: 10.11648/j.acm.20251406.15

Copy | Download

@article{10.11648/j.acm.20251406.15,
  author = {Qiaoling Wang and Haizhen Ren},
  title = {On the Edge Connectivity of Semi-strong Product Graphs},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {6},
  pages = {356-359},
  doi = {10.11648/j.acm.20251406.15},
  url = {https://doi.org/10.11648/j.acm.20251406.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251406.15},
  abstract = {The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs  and , their semi-strong product  is a graph whose vertex set is , and the edge set  is defined as follows: if  and  are two vertices in the semi-strong product , then there is an edge between them if and only if  and  and  are adjacent in , or  and  are adjacent in  and  and  are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - On the Edge Connectivity of Semi-strong Product Graphs
AU  - Qiaoling Wang
AU  - Haizhen Ren
Y1  - 2025/12/11
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251406.15
DO  - 10.11648/j.acm.20251406.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 356
EP  - 359
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251406.15
AB  - The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs  and , their semi-strong product  is a graph whose vertex set is , and the edge set  is defined as follows: if  and  are two vertices in the semi-strong product , then there is an edge between them if and only if  and  and  are adjacent in , or  and  are adjacent in  and  and  are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.
VL  - 14
IS  - 6
ER  -

Copy | Download