A Fundamental Theorem on Graph Operators

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.