Extremal graphs with minimum number of connected subgraphs in a given family
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915435042045952 |
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| author | Pandey, Dinesh Ravi, Peruvemba Sundaram |
| author_facet | Pandey, Dinesh Ravi, Peruvemba Sundaram |
| contents | The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut vertices have been characterised. Further, using this characterisation, the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices, with girth at least $k$, have been obtained. This turns out to characterise the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices for $0 \leq k \leq 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06476 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal graphs with minimum number of connected subgraphs in a given family Pandey, Dinesh Ravi, Peruvemba Sundaram Combinatorics 05C30, 05C35, 05C75 The subgraph number of a vertex in a graph is defined as the number of connected subgraphs containing that vertex. The graph and its vertex which correspond to the minimum subgraph number among all graphs on $n$ vertices and $k$ cut vertices have been characterised. Further, using this characterisation, the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices, with girth at least $k$, have been obtained. This turns out to characterise the graphs with the minimum number of connected subgraphs among all graphs on $n$ vertices and $k$ cut vertices for $0 \leq k \leq 4$. |
| title | Extremal graphs with minimum number of connected subgraphs in a given family |
| topic | Combinatorics 05C30, 05C35, 05C75 |
| url | https://arxiv.org/abs/2508.06476 |