On Additively Geometric Graphs

Abstract

Let $N$ and $Z$ denote respectively the set of all nonnegative integers and the set of all integers. A $(p,q)$-graph $G=(V,E)$ is said to be additively $(a,r)$-geometric if there exists an injective function $f:V\to Z$ such that $f^{+}(E)=\{a,ar,\dots ,a{r}^{q-1}\}$ where $a,r\in N$, $r>1$, and $f^{+}$ is defined by $f^{+}(uv)=f(u)+f(v)$ for all $uv\in E$. If further $f(v)\in N$ for all $v\in V$, then $G$ is said to be additively $(a,r{)}^{\ast }$-geometric. In this paper we characterise graphs which are additively geometric and additively ${}^{\ast }$-geometric.