Bicyclic graphs with the smallest and largest numbers of connected sets

TL;DR

This paper investigates the number of connected induced subgraphs in bicyclic graphs, identifying those with the minimum, maximum, and second maximum counts, and computes the extreme values within this family.

Contribution

It characterizes the extremal structures of bicyclic graphs with respect to the number of connected induced subgraphs and determines their exact extreme values.

Findings

Identified structures with the smallest and largest N(G) in bicyclic graphs.

Determined the second-largest N(G) structures.

Computed the exact minimum and maximum N(G) values for n-vertex bicyclic graphs.

Abstract

For a graph $G$ with vertex set $V$, let N($G$) denote the number of nonempty subsets of $V$ that induce a connected graph in $G$. In this paper, we focus on determining N($G$) for $G$ in the family $\mathbb{B}_n$ of $n$-vertex bicyclic graphs. We find in $\mathbb{B}_n$ the structures of those graphs that possess the smallest, the largest, as well as the second-largest values of N($G$). Moreover, we compute the extreme values of N($G$) over $\mathbb{B}_n$.