  The inertia of a graph $G$ is defined to be the triplet $In(G) = (p(G), n(G),
$ $\eta(G))$, where $p(G)$, $n(G)$ and $\eta(G)$ are the numbers of positive,
negative and zero eigenvalues (including multiplicities) of the adjacency
matrix $A(G)$, respectively. Traditionally $p(G)$ (resp. $n(G)$) is called the
positive (resp. negative) inertia index of $G$. In this paper, we introduce
three types of congruent transformations for graphs that keep the positive
inertia index and negative inertia index. By using these congruent
transformations, we determine all graphs with exactly two positive eigenvalues
and one zero eigenvalue.
