A Fundamental Theorem on Graph Operators

TL;DR

This paper establishes a fundamental classification of the long-term behavior of graph operators, showing they exhibit only three possible types of iterative behavior, and illustrates this with new specific operators.

Contribution

It proves a universal theorem characterizing the possible behaviors of graph operators under iteration, introducing new operators as examples.

Findings

Sequences either become empty, grow infinitely, or become periodic.

Classifies all possible asymptotic behaviors of graph operators.

Introduces the path and claw graph operators as examples.

Abstract

A graph operator is a function $\Gamma$ defined on some set of graphs such that whenever two graphs $G$ and $H$ are isomorphic, written $G\simeq H$, then $\Gamma(G)\simeq \Gamma(H)$. For a graph $G$ not in the domain of $\Gamma$, we put $\Gamma(G)=\emptyset$. Also, let us define $\Gamma^0(G)=G$, and for any integr $k\ge1$, $\Gamma^k(G)=\Gamma(\Gamma^{k-1}(G))$ We prove that if $\Gamma$ is a graph operator, then the sequence $\langle \Gamma^k(G)\rangle_{k=0}^\infty$ has only three possible types of behaviour. Either $\Gamma^k(G)=\emptyset$ for some integer $k>0$, or $\displaystyle\lim_{k\to\infty}|V(\Gamma^k(G))|=\infty$, or there exist integers $m\ge0$, $p>0$ such that the graphs $\Gamma^j(G)$ are non-isomorphic ($0\le j\le m)$, and $\Gamma^{n+p}\simeq \Gamma ^n(G)$ for all integers $n\ge m$. We illustrate this using two new graph operators, namely, the path graph operator and the claw graph operator.