The generalized 3-connectivity of a family regular networks
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909207909892096 |
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| author | Wang, Jing Luan, Xidao Huang, Yuanqiu |
| author_facet | Wang, Jing Luan, Xidao Huang, Yuanqiu |
| contents | The generalized $k$-connectivity of a graph $G$, denoted by $κ_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ with $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. In this paper, we firstly introduce a family of regular networks $H_n$ that can be obtained from several subgraphs $G_n^1, G_n^2, \cdots, G_n^{t_n}$ by adding a matching, where each subgraph $G_n^i$ is isomorphic to a particular graph $G_n$ ($1\le i\le t_n$). Then we determine the generalized 3-connectivity of $H_n$. As applications of the main result, the generalized 3-connectivity of some two-level interconnection networks, such as the hierarchical star graph $HS_n$, the hierarchical cubic network $HCN_n$ and the hierarchical folded hypercube $HFQ_n$, are determined directly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_00320 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The generalized 3-connectivity of a family regular networks Wang, Jing Luan, Xidao Huang, Yuanqiu Combinatorics The generalized $k$-connectivity of a graph $G$, denoted by $κ_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ with $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. In this paper, we firstly introduce a family of regular networks $H_n$ that can be obtained from several subgraphs $G_n^1, G_n^2, \cdots, G_n^{t_n}$ by adding a matching, where each subgraph $G_n^i$ is isomorphic to a particular graph $G_n$ ($1\le i\le t_n$). Then we determine the generalized 3-connectivity of $H_n$. As applications of the main result, the generalized 3-connectivity of some two-level interconnection networks, such as the hierarchical star graph $HS_n$, the hierarchical cubic network $HCN_n$ and the hierarchical folded hypercube $HFQ_n$, are determined directly. |
| title | The generalized 3-connectivity of a family regular networks |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.00320 |