plotting - Converting Mesh in DensityPlot to a Graph

Consider the following line of code:

DensityPlot[4 Sin[2 Pi x] Cos[1.5 Pi y] (1 - x^2) (1 - y) y, 
                  {x, -1, 1}, {y, 0, 1}, Mesh -> All, MeshStyle -> Thick]

with the following output:

How can I convert the shown mesh into a Graph object such that:

1. 
   

   All the vertices are aligned at the intersection of the lines, with correct VertexCoordinates.

   
   
   


2. 
   

   The edges (line segments in the plot) connect the corresponding vertices.

   
   

Note: If g is the resulting Graph, then the following code should give the same figure as the one above:

Show[DensityPlot[4 Sin[2 Pi x] Cos[1.5 Pi y] (1 - x^2) (1 - y) y, {x, -1, 1}, {y, 0, 1}], g]

Answer

This will do

densPlot = 

  DensityPlot[
   4 Sin[2 Pi x] Cos[1.5 Pi y] (1 - x^2) (1 - y) y, {x, -1, 1}, {y, 0,
     1}, MeshStyle -> Thick, Mesh -> All];

vertexCoordinates = densPlot[[1, 1]];

length = Length[vertexCoordinates];

graphReadyConnections =
  DeleteDuplicates@

   Flatten[
    Cases[#, 
       List[x_, y_, z_] :> {Sort[x \[UndirectedEdge] y], 
         Sort[x \[UndirectedEdge] z], Sort[y \[UndirectedEdge] z]}, 
       Infinity] &@
     densPlot[[1, 2, 1, 1, 3, 1, 1, 1]]
    ,
    1
    ];


Graph[Range[length], graphReadyConnections, 
 VertexCoordinates -> vertexCoordinates, 
 VertexShapeFunction -> {Disk[#, 0.005] &}, ImageSize -> 800]