Symmetry and asymmetry between positive and negative square energies of graphs

The positive and negative square energies of a graph, $s^{+}(G)$ and $s^{-}(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry between $s^{+}(G)$ and $s^{-}(G)$. This paper reviews examples of asymmetry between these parameters, for example using large random graphs and the ratios $s^{+}\mathrm{/}{s}^{-}$ and $s^{-}\mathrm{/}{s}^{+}$, as well as new examples of symmetry. We answer some questions previously asked about $s^{+}$ and $s^{-}$ and suggest several further avenues of research.