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The path minimises the average size of a connected induced subgraph

John Haslegrave

Abstract

We prove that among all graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter. While this paper was being prepared, a different proof was given by Andrew Vince.

The path minimises the average size of a connected induced subgraph

Abstract

We prove that among all graphs of order n, the path uniquely minimises the average order of its connected induced subgraphs. This confirms a conjecture of Kroeker, Mol and Oellermann, and generalises a classical result of Jamison for trees, as well as giving a new, shorter proof of the latter. While this paper was being prepared, a different proof was given by Andrew Vince.

Paper Structure

This paper contains 3 sections, 3 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof
  3. Final remarks

Key Result

Theorem 1

Let $G$ be a connected graph of order $n$. Then $A(G)\geq(n+2)/3$, with equality if and only if $G$ is a path.

Figures (1)

  • Figure 1: A graph $G$ with path $P=v_0v_1v_2$ (left) and a component of the auxiliary digraph $H$ (right). For the set $\{a\}$ at distance $2$ from $P$, the vertex $b$ was chosen. Double-headed arrows indicate blue edges. The set $\{a,b,v_0\}$ is a blue top but not a red top.

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 2
  • proof
  • Claim 2.1
  • proof : Proof of claim
  • Lemma 3
  • proof
  • Claim 3.1
  • proof : Proof of claim
  • Claim 3.2
  • ...and 4 more