On the existence of neutral graph

TL;DR

This paper investigates the existence of neutral graphs with zero assortativity coefficient, proving that for sufficiently large n, such graphs exist as trees and non-trees, addressing a longstanding conjecture.

Contribution

It demonstrates the existence of neutral graphs on n vertices for various n, including trees for n≥7 and non-trees for n≥13, advancing understanding of graph assortativity.

Findings

Neutral trees exist for n≥7.

Neutral non-tree graphs exist for n≥13.

Addresses the conjecture on the existence of neutral graphs.

Abstract

Graph is considered neutral if its assortativity coefficient $r$ is equal to zero. In this paper, we address an outstanding conjecture, i.e., whether is there a neutral graph on $n$ vertices? First, we show that for $n\geq7$, there is at least one neutral tree, which suggests that we find a representative of any order neutral graph. Additionally, we obtain that given $n\geq13$, there exists at least one neutral non-tree graph.