Here is the adjacency matrix of a graph with 60 nodes from a built-in MATLAB function.
[A,v] = bucky;
size(A)
The extra vector 
gives particular coordinates for each node on the unit sphere. Plotting the nodes and edges with these coordinates reveals beautiful structure.
gives particular coordinates for each node on the unit sphere. Plotting the nodes and edges with these coordinates reveals beautiful structure.gplot(A,v), axis equal
As you can see, the edges resemble those on a soccer ball. The same structure, when made of carbon atoms at the nodes, is called a buckyball in materials science.