Algebraic Distance Graphs and Rigidity

An algebraic distance graph is defined to be a graph with vertices in En in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that ( 1 ) if all the sides of a convex polygon T which is inscribed in a circle are algebraic numbers, then the circumradius and all diagonals of Y are also algebraic numbers, (2) the chromatic number of the algebraic distance graph on a circle of radius r is oo or 2 accordingly as r is algebraic or not. We also prove that for any n > 0, there exists a graph G which cannot be represented as an algebraic distance graph in En .