Equalizing edge lengths in simple graphs

I would like to generate a graph from an adjacency matrix in which each edge of the graph is forced to be the same length. The specific length of the edges is not important and could be any value (i.e. 1cm) as long as such a value results in a graph that can be easily understood, for example, edges that are too short may result in vertices being too close together and this may make the graph look "cluttered"

All input matrices are simple, square adjacency matrices and an example of one (19 x 19) is as follows:

A = [0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]

The idea here is that certain graphs (with a particular number of vertices of specific degree) cannot exist when all edges are forced to be exactly equal in length. Furthermore, certain graphs will be forced into specific geometries once all edges are forced to be equal in length. These are the things I am interested in.

This seems like a relatively simple process but graphs typically contain only topological properties (i.e. vertex connectivities) not geometrical properties such as edge lengths and thus I am not sure how to write code for this problem.

Please let me know if anyone has any ideas !

Thank you