A Fundamental Theorem on Graph Operators

Abstract

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