Construction of regular graphs What are the ways to create regular graphs using any number of vertices, under the below rule for adjacency between vertices.

Vertex A is adjacent to vertex B if and only if vertex B is within the fixed circular  range (i.e. within a given distance from vertex A) from vertex A. 

This circular range is fixed and equal for all the vertices. Vertices can be located in any way to create a regular graph. Please avoid mesh topologies (any vertex is adjacent to all other vertices in the graph), when creating regular graphs.
I created circular graphs and torus graphs under the above conditions, and would like to know whether there are any other methods to do this. 
 A: These are a special type of geometric graph; it looks like you're using the metric space $\mathbb{R}^{\text{something}}$ with the Euclidean distance, which are sometimes called space graphs.  These generalise to arbitrary metric spaces.  These are a special family of intersection graphs (including unit interval graphs).
The following theorem/proof is by Maehara (1984).

Theorem: Every $n$-vertex graph can be realised as a space graph in $\mathbb{R}^n$.

Proof: Suppose the vertices are $v_1,\ldots,v_n$ and let $A$ be the adjacency matrix of the graph (with row $i$ corresponding to vertex $v_i$).  We assign the vertex $v_i$ the point $\mathbf{x}_i \in \mathbb{R}^n$ whose coordinates are the $i$-th row of $A+nI$.
Hence $$||\mathbf{x}_i-\mathbf{x}_j||^2 \leq 2n^2-3n$$ if $i$ and $j$ are adjacent.  The maximum possible is when we have e.g.
$$\begin{array}{ccccccc}
\mathbf{x}_i= & (1 & 1 & n & 1 & 1 & \cdots & 1)\phantom{.} \\
\mathbf{x}_j= & (0 & 0 & 1 & n & 0 & \cdots & 0). \\
\end{array}$$
And $$||\mathbf{x}_i-\mathbf{x}_j||^2 \geq 2n^2$$ if $i$ and $j$ are not adjacent.  The minimum possible is when we have e.g.
$$\begin{array}{ccccccc}
\mathbf{x}_i= & (1 & 1 & n & 0 & 1 & \cdots & 1)\phantom{.} \\
\mathbf{x}_j= & (1 & 1 & 0 & n & 1 & \cdots & 1). \\
\end{array}$$
Hence by setting the boundary at a distance between these two extremal distances (followed by scaling, if we have a preferred boundary), we can realise the graph as a space graph.

H. Maehara, Space graphs and sphericity. Discrete Appl. Math. 7 (1984), no. 1, 55–64. 

