Generalized 4-connectivity of hierarchical star networks
The connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An SS-tree of a connected graph GG is a tree T=(V′,E′)T=\left(V^{\prime} ,E^{\prime} ) that contains all the vertices in SS subje...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-10-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2022-0490 |
| _version_ | 1811246836888895488 |
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| author | Wang Junzhen Zou Jinyu Zhang Shumin |
| author_facet | Wang Junzhen Zou Jinyu Zhang Shumin |
| author_sort | Wang Junzhen |
| collection | DOAJ |
| description | The connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An SS-tree of a connected graph GG is a tree T=(V′,E′)T=\left(V^{\prime} ,E^{\prime} ) that contains all the vertices in SS subject to S⊆V(G)S\subseteq V\left(G). Two SS-trees TT and T′T^{\prime} are internally disjoint if and only if E(T)∩E(T′)=∅E\left(T)\cap E\left(T^{\prime} )=\varnothing and V(T)∩V(T′)=SV\left(T)\cap V\left(T^{\prime} )=S. Denote by κ(S)\kappa \left(S) the maximum number of internally disjoint SS-trees in graph GG. The generalized kk-connectivity is defined as κk(G)=min{κ(S)∣S⊆V(G)and∣S∣=k}{\kappa }_{k}\left(G)=\min \left\{\kappa \left(S)| S\subseteq V\left(G)\hspace{0.33em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.33em}| S| \hspace{0.33em}=\hspace{0.33em}k\right\}. Clearly, κ2(G)=κ(G){\kappa }_{2}\left(G)=\kappa \left(G). In this article, we show that κ4(HSn)=n−1{\kappa }_{4}\left(H{S}_{n})=n-1, where HSnH{S}_{n} is the hierarchical star network. |
| first_indexed | 2024-04-12T14:59:09Z |
| format | Article |
| id | doaj.art-ae8c835b68d646f2ac664d6b2165b8c9 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-04-12T14:59:09Z |
| publishDate | 2022-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-ae8c835b68d646f2ac664d6b2165b8c92022-12-22T03:28:08ZengDe GruyterOpen Mathematics2391-54552022-10-012011261127510.1515/math-2022-0490Generalized 4-connectivity of hierarchical star networksWang Junzhen0Zou Jinyu1Zhang Shumin2School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810008, ChinaDepartment of Basic Research, Qinghai University, Xining, Qinghai 810008, ChinaSchool of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai 810008, ChinaThe connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An SS-tree of a connected graph GG is a tree T=(V′,E′)T=\left(V^{\prime} ,E^{\prime} ) that contains all the vertices in SS subject to S⊆V(G)S\subseteq V\left(G). Two SS-trees TT and T′T^{\prime} are internally disjoint if and only if E(T)∩E(T′)=∅E\left(T)\cap E\left(T^{\prime} )=\varnothing and V(T)∩V(T′)=SV\left(T)\cap V\left(T^{\prime} )=S. Denote by κ(S)\kappa \left(S) the maximum number of internally disjoint SS-trees in graph GG. The generalized kk-connectivity is defined as κk(G)=min{κ(S)∣S⊆V(G)and∣S∣=k}{\kappa }_{k}\left(G)=\min \left\{\kappa \left(S)| S\subseteq V\left(G)\hspace{0.33em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.33em}| S| \hspace{0.33em}=\hspace{0.33em}k\right\}. Clearly, κ2(G)=κ(G){\kappa }_{2}\left(G)=\kappa \left(G). In this article, we show that κ4(HSn)=n−1{\kappa }_{4}\left(H{S}_{n})=n-1, where HSnH{S}_{n} is the hierarchical star network.https://doi.org/10.1515/math-2022-0490hierarchical star networksfault-tolerancegeneralized connectivitydisjoint s-trees05c0505c4005c76 |
| spellingShingle | Wang Junzhen Zou Jinyu Zhang Shumin Generalized 4-connectivity of hierarchical star networks Open Mathematics hierarchical star networks fault-tolerance generalized connectivity disjoint s-trees 05c05 05c40 05c76 |
| title | Generalized 4-connectivity of hierarchical star networks |
| title_full | Generalized 4-connectivity of hierarchical star networks |
| title_fullStr | Generalized 4-connectivity of hierarchical star networks |
| title_full_unstemmed | Generalized 4-connectivity of hierarchical star networks |
| title_short | Generalized 4-connectivity of hierarchical star networks |
| title_sort | generalized 4 connectivity of hierarchical star networks |
| topic | hierarchical star networks fault-tolerance generalized connectivity disjoint s-trees 05c05 05c40 05c76 |
| url | https://doi.org/10.1515/math-2022-0490 |
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