On existence of neutral graph

TL;DR

The paper addresses whether neutral graphs with $r=0$ exist for arbitrary order and provides constructive methods to generate them. It introduces several graph-operations (subdivision, triangle, and leaf-connecting) and proves that neutral trees exist for $n\ge 7$, while neutral non-tree graphs exist for $n\ge 13$, with stronger constructions covering more orders via the $G^{\\oplus}$ framework. The results establish explicit neutral representatives of many orders and offer mechanisms to extend neutrality from trees to non-tree graphs. This advances understanding of degree assortativity in networks and provides practical tools for constructing neutral graphs of prescribed sizes.

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 exist at least one neutral non-tree graph.