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Bicyclic graphs with the smallest and largest numbers of connected sets

Audace A. V. Dossou-Olory

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 $\mathbb{B}_n$.

Bicyclic graphs with the smallest and largest numbers of connected sets

Abstract

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

Paper Structure

This paper contains 3 sections, 13 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The extremal graph structures

Key Result

Lemma 1

For the path $P_n$ and $u$ a leaf of $P_n$, we have

Figures (3)

  • Figure 4: All bicyclic graphs on five vertices. Their numbers of connected sets are $22,~24,~26,~23,~22$, respectively.
  • Figure 5: From Type II to a Type I.
  • Figure 6: The graphs $G,G',G"$ constructed in Lemma \ref{['singleBranch']}.

Theorems & Definitions (22)

  • Lemma 1: Wang2005
  • Lemma 2: Audacegenral2018AudaceGirth2018
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 12 more