Two notions of unit distance graphs

A faithful (unit) distance graph   in RdRd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph   in RdRd is any subgraph of such a graph.In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed d   the number of faithful distance graphs in RdRd on n   labelled vertices is 2(1+o(1))dnlog2n, and give a short proof of the known fact that the number of distance graphs in RdRd on n   labelled vertices is 2(1−1/⌊d/2⌋+o(1))n2/22(1−1/⌊d/2⌋+o(1))n2/2. We also study the behavior of several Ramsey-type quantities involving these graphs.In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in RdRd.