On the existence of neutral graph

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.