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

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 $0 \leq k \leq 4$.