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The inverse problem for the Steiner--Wiener index via additive number theory

Christian Bernert, Joshua Shaw

Abstract

We show that, for any given $k \ge 2$, every sufficiently large number appears as the Steiner--Wiener $k$ index of a graph.

The inverse problem for the Steiner--Wiener index via additive number theory

Abstract

We show that, for any given , every sufficiently large number appears as the Steiner--Wiener index of a graph.
Paper Structure (3 sections, 4 theorems, 26 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The graph-theoretic construction
  3. Counting solutions to diophantine equations

Key Result

Theorem 1

Let $k \ge 2$ be a positive integer. Then for all but finitely many positive integers $n$, there is a graph $G$ with $\mathrm{SW}_k(G)=n$.

Figures (1)

  • Figure 1: The nested star construction for $n=12, r=2$ with $a_1=3, a_2=8$

Theorems & Definitions (9)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Proposition \ref{['prop:graph']}
  • Remark
  • Proposition 4
  • proof : Proof of Proposition \ref{['prop:ant']}
  • proof : Proof of Proposition \ref{['prop:counting']}