  The energy of a graph $G$ is the sum of the absolute values of the
eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum
of the squares of the positive and negative eigenvalues of $G$, respectively.
It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math.
(2016)] that if $G$ is a connected graph of order $n$, then $s^+(G)\geq n-1$
and $s^-(G) \geq n-1$. In this paper, we show partial results towards this
conjecture. In particular, numerous structural results that may help in proving
the conjecture are derived, including the effect of various graph operations.
These are then used to establish the conjecture for several graph classes,
including graphs with certain fraction of positive eigenvalues and unicyclic
graphs.
