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arXiv · 2603.26812

Bicyclic graphs with the smallest and largest numbers of connected sets

Audace A. V. Dossou-Olory
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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 $\mathbb{B}_n$.

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Audace A. V. Dossou-Olory. 2026-03-26. Bicyclic graphs with the smallest and largest numbers of connected sets. https://arxiv.org/abs/2603.26812

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