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arXiv · 2508.06476

Extremal graphs with minimum number of connected subgraphs in a given family

Dinesh Pandey·Peruvemba Sundaram Ravi
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Abstract

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 $0 \leq k \leq 4$.

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Dinesh Pandey, Peruvemba Sundaram Ravi. 2025-08-08. Extremal graphs with minimum number of connected subgraphs in a given family. https://arxiv.org/abs/2508.06476

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