DISTANCE SETS WITH DIAMETER GRAPH BEING CYCLE

Abstract

A point set $X$ in the plane is called a $k$-distance set if there are exactly $k$ different distances between two distinct points in $X$. Let $D=D(X) $ be the diameter of a finite set $X$, and let $X_{D} = \{x\in X : d(x, y) = D$ for some $y \in X\}$, the diameter graph $DG(X_{D})$ of $X_{D}$ is the graph with $X_{D}$ as its vertices and where two vertices $x, y \in X_{D} $ are adjacent if $d(x, y) = D$. We prove the set $X$ having at most five distances with $DG(X_{D})=C_{7}$ has the unique $X_{D}=R_{7}$, and the set $X$ having at most six distances with $DG(X_{D})=C_{9}$ has the unique $X_{D}=R_{9}$, and give a conjecture for $k$-distance set with $DG(X_{D})=C_{2k-3}$.