# Spatial embedding of graph

Given a graph $$(V,E)$$, I'm interested in embedding it into a Euclidean space $$\mathbb{R}^n$$ such that each vertex $$v\in V$$ becomes a point $$x_v\in\mathbb{R}^n$$ and $$d(x_v,x_u) \leq 1$$ (Euclidean distance) iff $$(v,u) \in E$$.

Is this embedding always possible given high enough dimension $$n$$? If so, given a graph, can we estimate the upper bound for $$n$$ (as a function of $$|V|$$, or something similar) such that the above embedding is possible?

Your parameter is known as sphericity, first defined by Maehara, Space graphs and sphericity.

Maehara showed that every graph has such an embedding. Given a graph $$G = (V,E)$$, embed $$x \in V$$ into the $$V$$-indexed vector $$v_x$$ given by $$v_x(x) = M$$, $$v_x(y) = 1$$ if $$(x,y) \in E$$, and $$v_x(y) = 0$$ if $$(x,y) \notin E$$.

If $$(x,y) \notin E$$ then $$\|v_x - v_y\|^2 \geq 2M^2$$, considering coordinates $$x,y$$.

In contrast, if $$(x,y) \in E$$ then $$\|v_x - v_y\|^2 \leq 2(M-1)^2 + |V|-2$$.

When $$M$$ is large enough, $$2M^2 > 2(M-1)^2 + |V|-2$$, and so we can normalize the vectors so that they satisfy your condition.

The minimum dimension needed to embed a graph is known as its sphericity.

Maehara, Dispersed points and geometric embedding of complete bipartite graphs later showed that the sphericity of the complete bipartite graph $$K_{n,n}$$ is between $$n$$ and $$\tfrac32 n$$.

Some other results are cited in Bilu and Linial, Monotone maps, sphericity and bounded second eigenvalue.