is there a mathematical relation between percent overlap and window length or amplitude I am trying to plot short time energy of an utterance with percent overlap as an input variable.

As I referred to the short time energy equation it can be represented as square summation of $x(n)*w(n-m)$ where $w(n)$ seems to be window length.
But I couldn't figure out how to write it as percent overlap.
Percent overlap merely sets how often you measure the energy, relative to the window width.  You can re-measure every sample, or once per annum, or any density in between.