A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^{+}(G)$ and $s^{-}(G)$ is greater than $n$. More precisely, if $k\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, then $s^{+}(G)>n>s^{-}(G)$; if $k\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, then $s^{+}(G)<n<s^{-}(G)$. We confirm this conjecture.