  Let $(G,w)$ be a weighted graph with a weight-function $w: E(G)\to \mathbb
R\backslash\{0\}$. A weighted graph $(G,w)$ is invertible to a new weighted
graph if its adjacency matrix is invertible. A graph inverse has combinatorial
interest and can be applied to bound median eigenvalues of a graph such as have
physical meanings in Quatumn Chemistry. In this paper, we characterize the
inverse of a weighted graph based on its Sachs subgraphs that are spanning
subgraphs with only $K_2$ or cycles (or loops) as components. The
characterization can be used to find the inverse of a weighted graph based on
its structures instead of its adjacency matrix. If a graph has its spectra
split about the origin, i.e., half of eigenvalues are positive and half of them
are negative, then its median eigenvalues can be bounded by estimating the
largest and smallest eigenvalues of its inverse. We characterize graphs with a
unique Sachs subgraph and prove that these graphs has their spectra split about
the origin if they have a perfect matching. As applications, we show that the
median eigenvalues of stellated graphs of trees and corona graphs belong to
different halves of the interval $[-1,1]$.
