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

TL;DR

This paper characterizes extremal graphs with the fewest connected subgraphs given constraints on vertices, cut vertices, and girth, providing exact structures for small values of cut vertices.

Contribution

It introduces a characterization of graphs minimizing connected subgraphs with specified cut vertices and girth constraints, extending to small cut vertex counts.

Findings

Graphs with minimum connected subgraphs are characterized for k=0 to 4.

Explicit structures are provided for graphs with girth at least k.

The results generalize previous characterizations for specific graph classes.

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 $0 \leq k \leq 4$.