how to define the vertices and indices of a tesseract before and after projection into 3D ,is the way in which vertices are connected to form lines " Wireframe " in 4D remains the same after projection.
I'm learning graphics programming and my mathematical understanding isn't that much.
Suppose we think of a projection as a continuous mapping from $\mathbb{R}^4$ to $\mathbb{R}^3$. There are many kinds of projections used in computer graphics, but the computations generally represent continuous functions, sometimes affine or even linear.
In any case continuous linear mappings (which is a bit redundant; all finite dimensional linear maps are continuous) preserve the connectedness of vertices by their edges (linearity or affineness implying straight line segments in 4D go over to straight line segments in 3D).
As far as setting up a tesseract in 4D, the standard "unit" tesseract would be sixteen vertices with four 0,1 coordinates and edges between any pair of points whose coordinates are the same in all but one coordinate. You can scale this up to a larger size if that tesseract is too small for graphic display. The vertex labelling might conveniently be done with numbers 0-15 by identifying a binary representation with their coordinates.